TPTP Problem File: SLH0553^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0014_IKKBZ_Examples/prob_00091_003527__16016980_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1620 ( 592 unt; 337 typ; 0 def)
% Number of atoms : 3741 (1706 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 13264 ( 537 ~; 46 |; 504 &;10819 @)
% ( 0 <=>;1358 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 848 ( 848 >; 0 *; 0 +; 0 <<)
% Number of symbols : 302 ( 299 usr; 18 con; 0-5 aty)
% Number of variables : 3753 ( 187 ^;3169 !; 397 ?;3753 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:08:38.963
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
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thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_T,type,
t: pre_pr7278220950009878019t_unit ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1277)
thf(fact_0_forward__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_arcs_alt
thf(fact_1_hd__last__not__fwd__arcs,axiom,
! [X: a,Xs: list_a] :
~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ) ).
% hd_last_not_fwd_arcs
thf(fact_2_forward__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) ) ) ) ).
% forward_arcs.cases
thf(fact_3_no__back__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [X2: a,Xs2: list_a] :
( X
!= ( cons_a @ X2 @ Xs2 ) ) ) ).
% no_back_arcs.cases
thf(fact_4_forward__split,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs ) ) ).
% forward_split
thf(fact_5_forward__single,axiom,
! [X: a] : ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_single
thf(fact_6_forward__arcs__split,axiom,
! [Ys: list_a,Xs: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( append_a @ Ys @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_split
thf(fact_7_not__fwd__if__skip1,axiom,
! [Y: a,X: a,X3: a,Xs: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ ( cons_a @ X @ ( cons_a @ X3 @ Xs ) ) ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ ( cons_a @ X3 @ Xs ) ) )
=> ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ ( cons_a @ X3 @ Xs ) ) ) ) ) ).
% not_fwd_if_skip1
thf(fact_8_two__elems__if__not__fwd__conc,axiom,
! [Y: a,Xs: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) )
=> ? [A: a,B: a,Cs: list_a] :
( ( cons_a @ A @ ( cons_a @ B @ Cs ) )
= ( cons_a @ Y @ Xs ) ) ) ).
% two_elems_if_not_fwd_conc
thf(fact_9_forward__arcs_Osimps_I1_J,axiom,
iKKBZ_4180558001818622352cs_a_b @ t @ nil_a ).
% forward_arcs.simps(1)
thf(fact_10_forward__cons,axiom,
! [X: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_cons
thf(fact_11_forward__arcs_Osimps_I2_J,axiom,
! [X: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_arcs.simps(2)
thf(fact_12_forward__arcs__single,axiom,
! [X: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_arcs_single
thf(fact_13_nempty__if__not__fwd__conc,axiom,
! [Y: a,Xs: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) )
=> ( Xs != nil_a ) ) ).
% nempty_if_not_fwd_conc
thf(fact_14_hd__not__fwd__arcs,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( append_a @ Ys @ ( cons_a @ X @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ) ) ).
% hd_not_fwd_arcs
thf(fact_15_forward__arcs__alt_H,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt'
thf(fact_16_forward__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt_aux2
thf(fact_17_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_18_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_19_source__nmem__k__nh,axiom,
! [V2: a,W: b > real,K: real] :
~ ( member_a @ V2 @ ( graph_3921080825633621230od_a_b @ t @ W @ V2 @ K ) ) ).
% source_nmem_k_nh
thf(fact_20_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_21_append1__eq__conv,axiom,
! [Xs: list_b,X: b,Ys: list_b,Y: b] :
( ( ( append_b @ Xs @ ( cons_b @ X @ nil_b ) )
= ( append_b @ Ys @ ( cons_b @ Y @ nil_b ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_22_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_23_append_Oright__neutral,axiom,
! [A2: list_b] :
( ( append_b @ A2 @ nil_b )
= A2 ) ).
% append.right_neutral
thf(fact_24_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_25_append__Nil2,axiom,
! [Xs: list_b] :
( ( append_b @ Xs @ nil_b )
= Xs ) ).
% append_Nil2
thf(fact_26_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_27_append__self__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_b ) ) ).
% append_self_conv
thf(fact_28_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_29_self__append__conv,axiom,
! [Y: list_b,Ys: list_b] :
( ( Y
= ( append_b @ Y @ Ys ) )
= ( Ys = nil_b ) ) ).
% self_append_conv
thf(fact_30_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_31_append__self__conv2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_b ) ) ).
% append_self_conv2
thf(fact_32_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_33_self__append__conv2,axiom,
! [Y: list_b,Xs: list_b] :
( ( Y
= ( append_b @ Xs @ Y ) )
= ( Xs = nil_b ) ) ).
% self_append_conv2
thf(fact_34_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_35_Nil__is__append__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( nil_b
= ( append_b @ Xs @ Ys ) )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% Nil_is_append_conv
thf(fact_36_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_37_append__is__Nil__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= nil_b )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% append_is_Nil_conv
thf(fact_38_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_39_list_Oinject,axiom,
! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
( ( ( cons_b @ X21 @ X22 )
= ( cons_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_40_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_41_same__append__eq,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_42_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_43_append__same__eq,axiom,
! [Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( append_b @ Ys @ Xs )
= ( append_b @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_44_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_45_append__assoc,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( append_b @ ( append_b @ Xs @ Ys ) @ Zs )
= ( append_b @ Xs @ ( append_b @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_46_append_Oassoc,axiom,
! [A2: list_a,B2: list_a,C: list_a] :
( ( append_a @ ( append_a @ A2 @ B2 ) @ C )
= ( append_a @ A2 @ ( append_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_47_append_Oassoc,axiom,
! [A2: list_b,B2: list_b,C: list_b] :
( ( append_b @ ( append_b @ A2 @ B2 ) @ C )
= ( append_b @ A2 @ ( append_b @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_48_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_49_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_50_rev__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_51_rev__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( rev_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( rev_b @ Ys ) @ ( rev_b @ Xs ) ) ) ).
% rev_append
thf(fact_52_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_53_rev__is__Nil__conv,axiom,
! [Xs: list_b] :
( ( ( rev_b @ Xs )
= nil_b )
= ( Xs = nil_b ) ) ).
% rev_is_Nil_conv
thf(fact_54_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_55_Nil__is__rev__conv,axiom,
! [Xs: list_b] :
( ( nil_b
= ( rev_b @ Xs ) )
= ( Xs = nil_b ) ) ).
% Nil_is_rev_conv
thf(fact_56_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_57_singleton__rev__conv,axiom,
! [X: b,Xs: list_b] :
( ( ( cons_b @ X @ nil_b )
= ( rev_b @ Xs ) )
= ( ( cons_b @ X @ nil_b )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_58_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_59_rev__singleton__conv,axiom,
! [Xs: list_b,X: b] :
( ( ( rev_b @ Xs )
= ( cons_b @ X @ nil_b ) )
= ( Xs
= ( cons_b @ X @ nil_b ) ) ) ).
% rev_singleton_conv
thf(fact_60_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_61_rev__eq__Cons__iff,axiom,
! [Xs: list_b,Y: b,Ys: list_b] :
( ( ( rev_b @ Xs )
= ( cons_b @ Y @ Ys ) )
= ( Xs
= ( append_b @ ( rev_b @ Ys ) @ ( cons_b @ Y @ nil_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_62_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_63_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A3: set_b] :
( ( collect_b
@ ^ [X4: b] : ( member_b @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_73_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_74_rev_Osimps_I1_J,axiom,
( ( rev_b @ nil_b )
= nil_b ) ).
% rev.simps(1)
thf(fact_75_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_76_not__Cons__self2,axiom,
! [X: b,Xs: list_b] :
( ( cons_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_77_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_78_append__eq__append__conv2,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,Ts: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Ts ) )
= ( ? [Us: list_b] :
( ( ( Xs
= ( append_b @ Zs @ Us ) )
& ( ( append_b @ Us @ Ys )
= Ts ) )
| ( ( ( append_b @ Xs @ Us )
= Zs )
& ( Ys
= ( append_b @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_79_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_80_append__eq__appendI,axiom,
! [Xs: list_b,Xs1: list_b,Zs: list_b,Ys: list_b,Us2: list_b] :
( ( ( append_b @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_b @ Xs1 @ Us2 ) )
=> ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_81_rev_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_82_rev_Osimps_I2_J,axiom,
! [X: b,Xs: list_b] :
( ( rev_b @ ( cons_b @ X @ Xs ) )
= ( append_b @ ( rev_b @ Xs ) @ ( cons_b @ X @ nil_b ) ) ) ).
% rev.simps(2)
thf(fact_83_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_84_Cons__eq__appendI,axiom,
! [X: b,Xs1: list_b,Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( cons_b @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_b @ Xs1 @ Zs ) )
=> ( ( cons_b @ X @ Xs )
= ( append_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_85_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_86_append__Cons,axiom,
! [X: b,Xs: list_b,Ys: list_b] :
( ( append_b @ ( cons_b @ X @ Xs ) @ Ys )
= ( cons_b @ X @ ( append_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_87_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_88_list__nonempty__induct,axiom,
! [Xs: list_b,P: list_b > $o] :
( ( Xs != nil_b )
=> ( ! [X2: b] : ( P @ ( cons_b @ X2 @ nil_b ) )
=> ( ! [X2: b,Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_b @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_89_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_90_list__induct2_H,axiom,
! [P: list_a > list_b > $o,Xs: list_a,Ys: list_b] :
( ( P @ nil_a @ nil_b )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_b )
=> ( ! [Y2: b,Ys2: list_b] : ( P @ nil_a @ ( cons_b @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_91_list__induct2_H,axiom,
! [P: list_b > list_a > $o,Xs: list_b,Ys: list_a] :
( ( P @ nil_b @ nil_a )
=> ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_b @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_92_list__induct2_H,axiom,
! [P: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
( ( P @ nil_b @ nil_b )
=> ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_b )
=> ( ! [Y2: b,Ys2: list_b] : ( P @ nil_b @ ( cons_b @ Y2 @ Ys2 ) )
=> ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_93_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_94_neq__Nil__conv,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
= ( ? [Y3: b,Ys3: list_b] :
( Xs
= ( cons_b @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_95_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_96_transpose_Ocases,axiom,
! [X: list_list_b] :
( ( X != nil_list_b )
=> ( ! [Xss: list_list_b] :
( X
!= ( cons_list_b @ nil_b @ Xss ) )
=> ~ ! [X2: b,Xs2: list_b,Xss: list_list_b] :
( X
!= ( cons_list_b @ ( cons_b @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_97_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_98_list_Oexhaust,axiom,
! [Y: list_b] :
( ( Y != nil_b )
=> ~ ! [X212: b,X222: list_b] :
( Y
!= ( cons_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_99_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_100_list_OdiscI,axiom,
! [List: list_b,X21: b,X22: list_b] :
( ( List
= ( cons_b @ X21 @ X22 ) )
=> ( List != nil_b ) ) ).
% list.discI
thf(fact_101_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_102_list_Odistinct_I1_J,axiom,
! [X21: b,X22: list_b] :
( nil_b
!= ( cons_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_103_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_104_eq__Nil__appendI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_b @ nil_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_105_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_106_append_Oleft__neutral,axiom,
! [A2: list_b] :
( ( append_b @ nil_b @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_107_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_108_append__Nil,axiom,
! [Ys: list_b] :
( ( append_b @ nil_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_109_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_110_rev__nonempty__induct,axiom,
! [Xs: list_b,P: list_b > $o] :
( ( Xs != nil_b )
=> ( ! [X2: b] : ( P @ ( cons_b @ X2 @ nil_b ) )
=> ( ! [X2: b,Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( ( P @ Xs2 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_111_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_112_append__eq__Cons__conv,axiom,
! [Ys: list_b,Zs: list_b,X: b,Xs: list_b] :
( ( ( append_b @ Ys @ Zs )
= ( cons_b @ X @ Xs ) )
= ( ( ( Ys = nil_b )
& ( Zs
= ( cons_b @ X @ Xs ) ) )
| ? [Ys4: list_b] :
( ( Ys
= ( cons_b @ X @ Ys4 ) )
& ( ( append_b @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_113_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_114_Cons__eq__append__conv,axiom,
! [X: b,Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( cons_b @ X @ Xs )
= ( append_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_b )
& ( ( cons_b @ X @ Xs )
= Zs ) )
| ? [Ys4: list_b] :
( ( ( cons_b @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_b @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_115_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_116_rev__exhaust,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ~ ! [Ys2: list_b,Y2: b] :
( Xs
!= ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ).
% rev_exhaust
thf(fact_117_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_118_rev__induct,axiom,
! [P: list_b > $o,Xs: list_b] :
( ( P @ nil_b )
=> ( ! [X2: b,Xs2: list_b] :
( ( P @ Xs2 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_119_cycle__free,axiom,
~ ? [X_1: list_b] : ( arc_pre_cycle_a_b @ t @ X_1 ) ).
% cycle_free
thf(fact_120_hd__not__y__if__if__nfwd__app__fwd,axiom,
! [Y: a,Xs: list_a,Ys: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) )
=> ( ( hd_a @ ( rev_a @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) ) )
!= Y ) ) ) ).
% hd_not_y_if_if_nfwd_app_fwd
thf(fact_121_no__back__arcs__single,axiom,
! [X: a] : ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% no_back_arcs_single
thf(fact_122_seq__conform__single,axiom,
! [X: a] : ( iKKBZ_4622586873178280511rm_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% seq_conform_single
thf(fact_123_move__mid__backward__if__noarc,axiom,
! [U: list_a,V3: list_a,As: list_a,Bs: list_a,Cs2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ U @ V3 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ Bs @ ( append_a @ V3 @ Cs2 ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ V3 @ ( append_a @ Bs @ Cs2 ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc
thf(fact_124_scc__of__eq,axiom,
! [U2: a,V2: a] :
( ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ t @ V2 ) )
=> ( ( digrap2937667069914300949of_a_b @ t @ U2 )
= ( digrap2937667069914300949of_a_b @ t @ V2 ) ) ) ).
% scc_of_eq
thf(fact_125_no__back__single,axiom,
! [X: a] : ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% no_back_single
thf(fact_126_no__back__arcs_Osimps_I1_J,axiom,
iKKBZ_7773321254043928001cs_a_b @ t @ nil_a ).
% no_back_arcs.simps(1)
thf(fact_127_before__forward2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 ) ) ).
% before_forward2I
thf(fact_128_before__forward1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 ) ) ).
% before_forward1I
thf(fact_129_awalk__verts_Osimps_I1_J,axiom,
! [U2: a] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ nil_b )
= ( cons_a @ U2 @ nil_a ) ) ).
% awalk_verts.simps(1)
thf(fact_130_awalk__verts__non__Nil,axiom,
! [U2: a,P2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
!= nil_a ) ).
% awalk_verts_non_Nil
thf(fact_131_awalk__verts__ne__eq,axiom,
! [P2: list_b,U2: a,V2: a] :
( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( arc_pr7493981781705774526ts_a_b @ t @ V2 @ P2 ) ) ) ).
% awalk_verts_ne_eq
thf(fact_132_awhd__append,axiom,
! [U2: a,P2: list_b,Q: list_b] :
( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) ) )
= ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q ) ) @ P2 ) ) ) ).
% awhd_append
thf(fact_133_no__back__insert,axiom,
! [X: a,Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ Xs ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ).
% no_back_insert
thf(fact_134_no__back__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
= ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt
thf(fact_135_no__back__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt_aux2
thf(fact_136_before__no__back1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S1 ) ) ).
% before_no_back1I
thf(fact_137_before__no__back2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S2 ) ) ).
% before_no_back2I
thf(fact_138_before__conform1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 ) ) ).
% before_conform1I
thf(fact_139_before__conform2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 ) ) ).
% before_conform2I
thf(fact_140_no__back__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% no_back_before
thf(fact_141_seq__conform__if__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% seq_conform_if_before
thf(fact_142_seq__conform__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
& ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% seq_conform_alt
thf(fact_143_seq__conform__def,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% seq_conform_def
thf(fact_144_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_145_hd__append2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ).
% hd_append2
thf(fact_146_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_147_list_Osel_I1_J,axiom,
! [X21: b,X22: list_b] :
( ( hd_b @ ( cons_b @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_148_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_149_hd__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( Xs = nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Ys ) ) )
& ( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ) ).
% hd_append
thf(fact_150_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_151_longest__common__prefix,axiom,
! [Xs: list_b,Ys: list_b] :
? [Ps: list_b,Xs3: list_b,Ys5: list_b] :
( ( Xs
= ( append_b @ Ps @ Xs3 ) )
& ( Ys
= ( append_b @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_b )
| ( Ys5 = nil_b )
| ( ( hd_b @ Xs3 )
!= ( hd_b @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_152_closed__w__imp__cycle,axiom,
! [P2: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P2 )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ t @ X_12 ) ) ).
% closed_w_imp_cycle
thf(fact_153_fwd__app__nhead__elem,axiom,
! [Xs: list_a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ( ( Y
!= ( hd_a @ Xs ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) ) ) ) ).
% fwd_app_nhead_elem
thf(fact_154_hd__in__verts__if__forward,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
=> ( member_a @ ( hd_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% hd_in_verts_if_forward
thf(fact_155_awalk__decomp__verts,axiom,
! [U2: a,P2: list_b,V2: a,Xs: list_a,Y: a,Ys: list_a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_cas_a_b @ t @ U2 @ Q2 @ Y )
=> ! [R: list_b] :
( ( arc_pre_cas_a_b @ t @ Y @ R @ V2 )
=> ( ( P2
= ( append_b @ Q2 @ R ) )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q2 )
= ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ Y @ R )
!= ( cons_a @ Y @ Ys ) ) ) ) ) ) ) ) ).
% awalk_decomp_verts
thf(fact_156_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr3327329314391289540t_unit,U2: a] :
( ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ nil_a )
= ( cons_a @ U2 @ nil_a ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_157_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr3994228789931197893t_unit,U2: b] :
( ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ nil_a )
= ( cons_b @ U2 @ nil_b ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_158_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr7945120425549786372t_unit,U2: b] :
( ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ nil_b )
= ( cons_b @ U2 @ nil_b ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_159_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ nil_b )
= ( cons_a @ U2 @ nil_a ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_160_seq__conform__if__dstnct__fwd,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs ) ) ) ).
% seq_conform_if_dstnct_fwd
thf(fact_161_no__back__if__distinct__forward,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% no_back_if_distinct_forward
thf(fact_162_awalk__verts__append3,axiom,
! [U2: a,P2: list_b,E: b,Q: list_b,R2: a,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( append_b @ P2 @ ( cons_b @ E @ Q ) ) @ R2 )
=> ( ( arc_pre_awalk_a_b @ t @ V2 @ Q @ R2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ ( cons_b @ E @ Q ) ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) @ ( arc_pr7493981781705774526ts_a_b @ t @ V2 @ Q ) ) ) ) ) ).
% awalk_verts_append3
thf(fact_163_awalk__verts__induce,axiom,
! [S: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) )
= ( arc_pr7493981781705774526ts_a_b @ t ) ) ).
% awalk_verts_induce
thf(fact_164_unique__awalk__All,axiom,
! [U2: a,V2: a] :
( ? [P3: list_b] : ( arc_pre_awalk_a_b @ t @ U2 @ P3 @ V2 )
=> ? [X2: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ X2 @ V2 )
& ! [Y4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ Y4 @ V2 )
=> ( Y4 = X2 ) ) ) ) ).
% unique_awalk_All
thf(fact_165_awalk__ends__eqD,axiom,
! [U2: a,P2: list_b,V2: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ U2 )
=> ( ( arc_pre_awalk_a_b @ t @ V2 @ P2 @ W )
=> ( V2 = W ) ) ) ).
% awalk_ends_eqD
thf(fact_166_distinct__mid__unique2,axiom,
! [Xs: list_a,U: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs @ ( append_a @ U @ Ys ) ) )
=> ( ( U != nil_a )
=> ( ( ( append_a @ Xs @ ( append_a @ U @ Ys ) )
= ( append_a @ As @ ( append_a @ U @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_167_distinct__mid__unique2,axiom,
! [Xs: list_b,U: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs @ ( append_b @ U @ Ys ) ) )
=> ( ( U != nil_b )
=> ( ( ( append_b @ Xs @ ( append_b @ U @ Ys ) )
= ( append_b @ As @ ( append_b @ U @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_168_distinct__mid__unique1,axiom,
! [Xs: list_a,U: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs @ ( append_a @ U @ Ys ) ) )
=> ( ( U != nil_a )
=> ( ( ( append_a @ Xs @ ( append_a @ U @ Ys ) )
= ( append_a @ As @ ( append_a @ U @ Bs ) ) )
=> ( As = Xs ) ) ) ) ).
% distinct_mid_unique1
thf(fact_169_distinct__mid__unique1,axiom,
! [Xs: list_b,U: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs @ ( append_b @ U @ Ys ) ) )
=> ( ( U != nil_b )
=> ( ( ( append_b @ Xs @ ( append_b @ U @ Ys ) )
= ( append_b @ As @ ( append_b @ U @ Bs ) ) )
=> ( As = Xs ) ) ) ) ).
% distinct_mid_unique1
thf(fact_170_awalk__last__in__verts,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_last_in_verts
thf(fact_171_awalk__hd__in__verts,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_hd_in_verts
thf(fact_172_awalk__appendI,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ V2 @ Q @ W )
=> ( arc_pre_awalk_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ W ) ) ) ).
% awalk_appendI
thf(fact_173_awalk__ends,axiom,
! [U2: a,P2: list_b,V2: a,U3: a,V4: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U3 @ P2 @ V4 )
=> ( ( ( P2 != nil_b )
& ( U2 = U3 )
& ( V2 = V4 ) )
| ( ( P2 = nil_b )
& ( U2 = V2 )
& ( U3 = V4 ) ) ) ) ) ).
% awalk_ends
thf(fact_174_awalk__empty__ends,axiom,
! [U2: a,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ nil_b @ V2 )
=> ( U2 = V2 ) ) ).
% awalk_empty_ends
thf(fact_175_cas__ends,axiom,
! [U2: a,P2: list_b,V2: a,U3: a,V4: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_cas_a_b @ t @ U3 @ P2 @ V4 )
=> ( ( ( P2 != nil_b )
& ( U2 = U3 )
& ( V2 = V4 ) )
| ( ( P2 = nil_b )
& ( U2 = V2 )
& ( U3 = V4 ) ) ) ) ) ).
% cas_ends
thf(fact_176_cas_Osimps_I1_J,axiom,
! [U2: a,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ nil_b @ V2 )
= ( U2 = V2 ) ) ).
% cas.simps(1)
thf(fact_177_in__scc__of__self,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ t @ U2 ) ) ) ).
% in_scc_of_self
thf(fact_178_hd__in__awalk__verts_I1_J,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( member_a @ U2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% hd_in_awalk_verts(1)
thf(fact_179_awalk__Nil__iff,axiom,
! [U2: a,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ nil_b @ V2 )
= ( ( U2 = V2 )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awalk_Nil_iff
thf(fact_180_awhd__if__cas,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= U2 ) ) ).
% awhd_if_cas
thf(fact_181_awalk__decomp,axiom,
! [U2: a,P2: list_b,V2: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ? [Q2: list_b,R: list_b] :
( ( P2
= ( append_b @ Q2 @ R ) )
& ( arc_pre_awalk_a_b @ t @ U2 @ Q2 @ W )
& ( arc_pre_awalk_a_b @ t @ W @ R @ V2 ) ) ) ) ).
% awalk_decomp
thf(fact_182_rotate__awalkE,axiom,
! [U2: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ U2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ~ ! [Q2: list_b,R: list_b] :
( ( P2
= ( append_b @ Q2 @ R ) )
=> ( ( arc_pre_awalk_a_b @ t @ W @ ( append_b @ R @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ) ).
% rotate_awalkE
thf(fact_183_awalk__verts__append__distinct,axiom,
! [R2: a,P1: list_b,P22: list_b] :
( ? [X_1: a] : ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) @ X_1 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) ) )
=> ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P1 ) ) ) ) ).
% awalk_verts_append_distinct
thf(fact_184_awalk__cyc__decompE_H,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
=> ~ ! [Q2: list_b,R: list_b,S3: list_b] :
( ( P2
= ( append_b @ Q2 @ ( append_b @ R @ S3 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q2 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ Q2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S3 @ V2 ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R ) ) ) ) ) ) ).
% awalk_cyc_decompE'
thf(fact_185_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_186_set__rev,axiom,
! [Xs: list_b] :
( ( set_b2 @ ( rev_b @ Xs ) )
= ( set_b2 @ Xs ) ) ).
% set_rev
thf(fact_187_set__rev,axiom,
! [Xs: list_list_a] :
( ( set_list_a2 @ ( rev_list_a @ Xs ) )
= ( set_list_a2 @ Xs ) ) ).
% set_rev
thf(fact_188_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_189_distinct__rev,axiom,
! [Xs: list_b] :
( ( distinct_b @ ( rev_b @ Xs ) )
= ( distinct_b @ Xs ) ) ).
% distinct_rev
thf(fact_190_awhd__of__awalk,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= U2 ) ) ).
% awhd_of_awalk
thf(fact_191_distinct_Osimps_I2_J,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) )
= ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_192_distinct_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X @ Xs ) )
= ( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_193_distinct_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X @ Xs ) )
= ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( distinct_list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_194_distinct_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_195_distinct_Osimps_I2_J,axiom,
! [X: b,Xs: list_b] :
( ( distinct_b @ ( cons_b @ X @ Xs ) )
= ( ~ ( member_b @ X @ ( set_b2 @ Xs ) )
& ( distinct_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_196_pre__digraph_Ocas_Ocong,axiom,
arc_pre_cas_a_b = arc_pre_cas_a_b ).
% pre_digraph.cas.cong
thf(fact_197_pre__digraph_Ocas_Ocong,axiom,
arc_pre_cas_list_a_b = arc_pre_cas_list_a_b ).
% pre_digraph.cas.cong
thf(fact_198_pre__digraph_Oawalk_Ocong,axiom,
arc_pre_awalk_a_b = arc_pre_awalk_a_b ).
% pre_digraph.awalk.cong
thf(fact_199_pre__digraph_Oawalk_Ocong,axiom,
arc_pr6214585750886380800st_a_b = arc_pr6214585750886380800st_a_b ).
% pre_digraph.awalk.cong
thf(fact_200_wf__digraph_Oclosed__w_Ocong,axiom,
arc_wf_closed_w_a_b = arc_wf_closed_w_a_b ).
% wf_digraph.closed_w.cong
thf(fact_201_pre__digraph_Ocas_Osimps_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V2: a] :
( ( arc_pre_cas_a_b @ G @ U2 @ nil_b @ V2 )
= ( U2 = V2 ) ) ).
% pre_digraph.cas.simps(1)
thf(fact_202_pre__digraph_Ocas_Osimps_I1_J,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a,V2: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U2 @ nil_b @ V2 )
= ( U2 = V2 ) ) ).
% pre_digraph.cas.simps(1)
thf(fact_203_not__distinct__conv__prefix,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( ~ ( distin132333870042060960od_a_a @ As ) )
= ( ? [Xs4: list_P1396940483166286381od_a_a,Y3: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Xs4 ) )
& ( distin132333870042060960od_a_a @ Xs4 )
& ( As
= ( append5335208819046833346od_a_a @ Xs4 @ ( cons_P7316939126706565853od_a_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_204_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs4: list_set_a,Y3: set_a,Ys3: list_set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Xs4 ) )
& ( distinct_set_a @ Xs4 )
& ( As
= ( append_set_a @ Xs4 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_205_not__distinct__conv__prefix,axiom,
! [As: list_list_a] :
( ( ~ ( distinct_list_a @ As ) )
= ( ? [Xs4: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs4 ) )
& ( distinct_list_a @ Xs4 )
& ( As
= ( append_list_a @ Xs4 @ ( cons_list_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_206_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs4: list_a,Y3: a,Ys3: list_a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs4 ) )
& ( distinct_a @ Xs4 )
& ( As
= ( append_a @ Xs4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_207_not__distinct__conv__prefix,axiom,
! [As: list_b] :
( ( ~ ( distinct_b @ As ) )
= ( ? [Xs4: list_b,Y3: b,Ys3: list_b] :
( ( member_b @ Y3 @ ( set_b2 @ Xs4 ) )
& ( distinct_b @ Xs4 )
& ( As
= ( append_b @ Xs4 @ ( cons_b @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_208_distinct__length__2__or__more,axiom,
! [A2: a,B2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A2 @ ( cons_a @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct_a @ ( cons_a @ A2 @ Xs ) )
& ( distinct_a @ ( cons_a @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_209_distinct__length__2__or__more,axiom,
! [A2: b,B2: b,Xs: list_b] :
( ( distinct_b @ ( cons_b @ A2 @ ( cons_b @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct_b @ ( cons_b @ A2 @ Xs ) )
& ( distinct_b @ ( cons_b @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_210_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_211_distinct_Osimps_I1_J,axiom,
distinct_b @ nil_b ).
% distinct.simps(1)
thf(fact_212_set__ConsD,axiom,
! [Y: product_prod_a_a,X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_213_set__ConsD,axiom,
! [Y: set_a,X: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_214_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_215_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_216_set__ConsD,axiom,
! [Y: b,X: b,Xs: list_b] :
( ( member_b @ Y @ ( set_b2 @ ( cons_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_b @ Y @ ( set_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_217_list_Oset__cases,axiom,
! [E: product_prod_a_a,A2: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A2 ) )
=> ( ! [Z2: list_P1396940483166286381od_a_a] :
( A2
!= ( cons_P7316939126706565853od_a_a @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_a_a,Z2: list_P1396940483166286381od_a_a] :
( ( A2
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z2 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_218_list_Oset__cases,axiom,
! [E: set_a,A2: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A2 ) )
=> ( ! [Z2: list_set_a] :
( A2
!= ( cons_set_a @ E @ Z2 ) )
=> ~ ! [Z1: set_a,Z2: list_set_a] :
( ( A2
= ( cons_set_a @ Z1 @ Z2 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_219_list_Oset__cases,axiom,
! [E: list_a,A2: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A2 ) )
=> ( ! [Z2: list_list_a] :
( A2
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A2
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_220_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z2: list_a] :
( A2
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_221_list_Oset__cases,axiom,
! [E: b,A2: list_b] :
( ( member_b @ E @ ( set_b2 @ A2 ) )
=> ( ! [Z2: list_b] :
( A2
!= ( cons_b @ E @ Z2 ) )
=> ~ ! [Z1: b,Z2: list_b] :
( ( A2
= ( cons_b @ Z1 @ Z2 ) )
=> ~ ( member_b @ E @ ( set_b2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_222_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_223_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_224_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_225_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_226_list_Oset__intros_I1_J,axiom,
! [X21: b,X22: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_227_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_228_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_229_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_230_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_231_list_Oset__intros_I2_J,axiom,
! [Y: b,X22: list_b,X21: b] :
( ( member_b @ Y @ ( set_b2 @ X22 ) )
=> ( member_b @ Y @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_232_distinct__singleton,axiom,
! [X: a] : ( distinct_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_singleton
thf(fact_233_distinct__singleton,axiom,
! [X: b] : ( distinct_b @ ( cons_b @ X @ nil_b ) ) ).
% distinct_singleton
thf(fact_234_split__list__first__prop__iff,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_235_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_236_split__list__first__prop__iff,axiom,
! [Xs: list_b,P: b > $o] :
( ( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_b,X4: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys3 @ ( cons_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: b] :
( ( member_b @ Y3 @ ( set_b2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_237_split__list__last__prop__iff,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_list_a,X4: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_238_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_239_split__list__last__prop__iff,axiom,
! [Xs: list_b,P: b > $o] :
( ( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_b,X4: b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys3 @ ( cons_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: b] :
( ( member_b @ Y3 @ ( set_b2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_240_in__set__conv__decomp__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_241_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_242_in__set__conv__decomp__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_243_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_244_in__set__conv__decomp__first,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_245_in__set__conv__decomp__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_246_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_247_in__set__conv__decomp__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_248_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_249_in__set__conv__decomp__last,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_250_split__list__first__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_251_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_252_split__list__first__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: b] :
( ( member_b @ Xa @ ( set_b2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_253_split__list__last__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X2: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_254_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_255_split__list__last__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_b,X2: b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: b] :
( ( member_b @ Xa @ ( set_b2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_256_split__list__first__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_257_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_258_split__list__first__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: b] :
( ( member_b @ Xa @ ( set_b2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_259_split__list__last__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X2: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_260_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_261_split__list__last__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_b,X2: b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: b] :
( ( member_b @ Xa @ ( set_b2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_262_in__set__conv__decomp,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_263_in__set__conv__decomp,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_264_in__set__conv__decomp,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_265_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_266_in__set__conv__decomp,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( Xs
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_267_append__Cons__eq__iff,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs5: list_P1396940483166286381od_a_a,Ys6: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs5 @ ( cons_P7316939126706565853od_a_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_268_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Xs5: list_set_a,Ys6: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ Ys ) )
= ( append_set_a @ Xs5 @ ( cons_set_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_269_append__Cons__eq__iff,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs5: list_list_a,Ys6: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
= ( append_list_a @ Xs5 @ ( cons_list_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_270_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys6: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_271_append__Cons__eq__iff,axiom,
! [X: b,Xs: list_b,Ys: list_b,Xs5: list_b,Ys6: list_b] :
( ~ ( member_b @ X @ ( set_b2 @ Xs ) )
=> ( ~ ( member_b @ X @ ( set_b2 @ Ys ) )
=> ( ( ( append_b @ Xs @ ( cons_b @ X @ Ys ) )
= ( append_b @ Xs5 @ ( cons_b @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_272_split__list__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_273_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_274_split__list__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_275_split__list__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_276_split__list__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_277_split__list__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_278_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_279_split__list__first,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs3 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_280_split__list__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_281_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_282_split__list__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_283_split__list__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_284_split__list__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_285_split__list__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_286_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_287_split__list__last,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs3 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_288_split__list,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_289_split__list,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_290_split__list,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_291_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_292_split__list,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs3: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_293_list_Oset__sel_I1_J,axiom,
! [A2: list_P1396940483166286381od_a_a] :
( ( A2 != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ A2 ) @ ( set_Product_prod_a_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_294_list_Oset__sel_I1_J,axiom,
! [A2: list_set_a] :
( ( A2 != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ A2 ) @ ( set_set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_295_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_296_list_Oset__sel_I1_J,axiom,
! [A2: list_b] :
( ( A2 != nil_b )
=> ( member_b @ ( hd_b @ A2 ) @ ( set_b2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_297_list_Oset__sel_I1_J,axiom,
! [A2: list_list_a] :
( ( A2 != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A2 ) @ ( set_list_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_298_hd__in__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_299_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_300_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_301_hd__in__set,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( member_b @ ( hd_b @ Xs ) @ ( set_b2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_302_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_303_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs3: list_a,Y2: a] :
( Ws
= ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_304_not__distinct__decomp,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs2: list_b,Ys2: list_b,Zs3: list_b,Y2: b] :
( Ws
= ( append_b @ Xs2 @ ( append_b @ ( cons_b @ Y2 @ nil_b ) @ ( append_b @ Ys2 @ ( append_b @ ( cons_b @ Y2 @ nil_b ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_305_pre__digraph_Oawalk__verts_Ocong,axiom,
arc_pr7493981781705774526ts_a_b = arc_pr7493981781705774526ts_a_b ).
% pre_digraph.awalk_verts.cong
thf(fact_306_directed__tree_Oforward_Ocong,axiom,
iKKBZ_4778857019735642799rd_a_b = iKKBZ_4778857019735642799rd_a_b ).
% directed_tree.forward.cong
thf(fact_307_directed__tree_Oforward__arcs_Ocong,axiom,
iKKBZ_4180558001818622352cs_a_b = iKKBZ_4180558001818622352cs_a_b ).
% directed_tree.forward_arcs.cong
thf(fact_308_directed__tree_Ono__back_Ocong,axiom,
iKKBZ_3684931046464919648ck_a_b = iKKBZ_3684931046464919648ck_a_b ).
% directed_tree.no_back.cong
thf(fact_309_directed__tree_Obefore_Ocong,axiom,
iKKBZ_7682935289300565975re_a_b = iKKBZ_7682935289300565975re_a_b ).
% directed_tree.before.cong
thf(fact_310_directed__tree_Oseq__conform_Ocong,axiom,
iKKBZ_4622586873178280511rm_a_b = iKKBZ_4622586873178280511rm_a_b ).
% directed_tree.seq_conform.cong
thf(fact_311_directed__tree_Ono__back__arcs_Ocong,axiom,
iKKBZ_7773321254043928001cs_a_b = iKKBZ_7773321254043928001cs_a_b ).
% directed_tree.no_back_arcs.cong
thf(fact_312_pre__digraph_Ocycle_Ocong,axiom,
arc_pre_cycle_a_b = arc_pre_cycle_a_b ).
% pre_digraph.cycle.cong
thf(fact_313_pre__digraph_Oawalk__verts__non__Nil,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,P2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 )
!= nil_a ) ).
% pre_digraph.awalk_verts_non_Nil
thf(fact_314_pre__digraph_Oawalk__verts__ne__eq,axiom,
! [P2: list_b,G: pre_pr7278220950009878019t_unit,U2: a,V2: a] :
( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 )
= ( arc_pr7493981781705774526ts_a_b @ G @ V2 @ P2 ) ) ) ).
% pre_digraph.awalk_verts_ne_eq
thf(fact_315_leaf__not__mem__awalk,axiom,
! [X: a,U2: a,P2: list_b,V2: a] :
( ( shorte1213025427933718126af_a_b @ t @ X )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( V2 != X )
=> ~ ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ).
% leaf_not_mem_awalk
thf(fact_316_mk__cycles__path__awalk,axiom,
! [U2: a,C: list_b,N: nat] :
( ( arc_pre_awalk_a_b @ t @ U2 @ C @ U2 )
=> ( arc_pre_awalk_a_b @ t @ U2 @ ( shorte6374615165232202367path_b @ N @ C ) @ U2 ) ) ).
% mk_cycles_path_awalk
thf(fact_317_awalk__del__vert,axiom,
! [U2: a,P2: list_b,V2: a,X: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ~ ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ t @ X ) @ U2 @ P2 @ V2 ) ) ) ).
% awalk_del_vert
thf(fact_318_rotate__trailE,axiom,
! [U2: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ U2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ~ ! [Q2: list_b,R: list_b] :
( ( P2
= ( append_b @ Q2 @ R ) )
=> ( ( arc_pre_trail_a_b @ t @ W @ ( append_b @ R @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE
thf(fact_319_in__set__inner__verts__appendI__r,axiom,
! [U2: a,Q: list_b,P2: list_b] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ Q ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_r
thf(fact_320_in__set__inner__verts__appendI__l,axiom,
! [U2: a,P2: list_b,Q: list_b] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_l
thf(fact_321_induce__subgraph__verts,axiom,
! [G: pre_pr2882871181989701257t_unit,Vs: set_list_a] :
( ( pre_ve1830060048215441954t_unit @ ( digrap21804061584661953st_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_322_induce__subgraph__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ve642382030648772252t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_323_hd__reach__all__if__nfwd__app__fwd,axiom,
! [Y: a,Xs: list_a,Ys: list_a,X: a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) )
=> ( ( member_a @ X @ ( set_a2 @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( rev_a @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) ) ) @ X ) ) ) ) ).
% hd_reach_all_if_nfwd_app_fwd
thf(fact_324_scc__of__in__sccs__verts,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ t @ U2 ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_325_reachable__trans,axiom,
! [U2: a,V2: a,W: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( reachable_a_b @ t @ V2 @ W )
=> ( reachable_a_b @ t @ U2 @ W ) ) ) ).
% reachable_trans
thf(fact_326_reachable__in__verts_I2_J,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(2)
thf(fact_327_reachable__in__verts_I1_J,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(1)
thf(fact_328_reachable__awalk,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
= ( ? [P4: list_b] : ( arc_pre_awalk_a_b @ t @ U2 @ P4 @ V2 ) ) ) ).
% reachable_awalk
thf(fact_329_reachable__awalkI,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( reachable_a_b @ t @ U2 @ V2 ) ) ).
% reachable_awalkI
thf(fact_330_trail__def,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ V2 )
= ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
& ( distinct_b @ P2 ) ) ) ).
% trail_def
thf(fact_331_k__nh__reachable,axiom,
! [U2: a,W: b > real,V2: a,K: real] :
( ( member_a @ U2 @ ( graph_3921080825633621230od_a_b @ t @ W @ V2 @ K ) )
=> ( reachable_a_b @ t @ V2 @ U2 ) ) ).
% k_nh_reachable
thf(fact_332_distinct__verts__imp__distinct,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
=> ( distinct_b @ P2 ) ) ) ).
% distinct_verts_imp_distinct
thf(fact_333_rotate__trailE_H,axiom,
! [U2: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ U2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_trail_a_b @ t @ W @ Q2 @ W )
=> ( ( ( set_b2 @ Q2 )
= ( set_b2 @ P2 ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ Q2 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE'
thf(fact_334_trail__Nil__iff,axiom,
! [U2: a,V2: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ nil_b @ V2 )
= ( ( U2 = V2 )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% trail_Nil_iff
thf(fact_335_awalk__verts__reachable__to,axiom,
! [U2: a,P2: list_b,V2: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( reachable_a_b @ t @ W @ V2 ) ) ) ).
% awalk_verts_reachable_to
thf(fact_336_awalk__verts__reachable__from,axiom,
! [U2: a,P2: list_b,V2: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( reachable_a_b @ t @ U2 @ W ) ) ) ).
% awalk_verts_reachable_from
thf(fact_337_hd__reach__all__forward,axiom,
! [Xs: list_a,X: a] :
( ( member_a @ ( hd_a @ Xs ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ Xs ) @ X ) ) ) ) ).
% hd_reach_all_forward
thf(fact_338_hd__reach__all__forward_H_H,axiom,
! [X: a,Y: a,Xs: list_a,Z: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
=> ( ( member_a @ Z @ ( set_a2 @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) ) @ Z ) ) ) ).
% hd_reach_all_forward''
thf(fact_339_hd__reach__all__forward__arcs,axiom,
! [Xs: list_a,X: a] :
( ( member_a @ ( hd_a @ ( rev_a @ Xs ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( rev_a @ Xs ) ) @ X ) ) ) ) ).
% hd_reach_all_forward_arcs
thf(fact_340_reachable__refl,axiom,
! [V2: a] :
( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V2 @ V2 ) ) ).
% reachable_refl
thf(fact_341_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ t @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_342_arc__balancedI__trail,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ V2 )
=> ( pre_ar5931435604406180204ed_a_b @ t @ U2 @ ( set_b2 @ P2 ) @ V2 ) ) ).
% arc_balancedI_trail
thf(fact_343_inner__verts__singleton,axiom,
! [X: b] :
( ( pre_inner_verts_a_b @ t @ ( cons_b @ X @ nil_b ) )
= nil_a ) ).
% inner_verts_singleton
thf(fact_344_pre__digraph_Otrail_Ocong,axiom,
arc_pre_trail_a_b = arc_pre_trail_a_b ).
% pre_digraph.trail.cong
thf(fact_345_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_346_pre__digraph_Otrail__def,axiom,
( arc_pr7309874995902050716st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P4: list_b,V5: list_a] :
( ( arc_pr6214585750886380800st_a_b @ G2 @ U4 @ P4 @ V5 )
& ( distinct_b @ P4 ) ) ) ) ).
% pre_digraph.trail_def
thf(fact_347_pre__digraph_Otrail__def,axiom,
( arc_pre_trail_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P4: list_b,V5: a] :
( ( arc_pre_awalk_a_b @ G2 @ U4 @ P4 @ V5 )
& ( distinct_b @ P4 ) ) ) ) ).
% pre_digraph.trail_def
thf(fact_348_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_349_del__vert__add__vert,axiom,
! [U2: a] :
( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ t @ U2 ) @ U2 )
= t ) ) ).
% del_vert_add_vert
thf(fact_350_inner__verts__Cons,axiom,
! [U2: a,E: b,Es: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ V2 )
=> ( ( ( Es != nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es ) )
= ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_inner_verts_a_b @ t @ Es ) ) ) )
& ( ( Es = nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es ) )
= nil_a ) ) ) ) ).
% inner_verts_Cons
thf(fact_351_cas__induce,axiom,
! [U2: a,P2: list_b,V2: a,S: set_a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ S )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U2 @ P2 @ V2 ) ) ) ).
% cas_induce
thf(fact_352_awalk__induce,axiom,
! [U2: a,P2: list_b,V2: a,S: set_a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ S )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U2 @ P2 @ V2 ) ) ) ).
% awalk_induce
thf(fact_353_merge__in__verts,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ t ) )
=> ( member_a @ X @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% merge_in_verts
thf(fact_354_to__list__tree__cas,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
= ( arc_pre_cas_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U2 @ nil_a ) @ P2 @ ( cons_a @ V2 @ nil_a ) ) ) ).
% to_list_tree_cas
thf(fact_355_to__list__tree__awalk,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
= ( arc_pr6214585750886380800st_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U2 @ nil_a ) @ P2 @ ( cons_a @ V2 @ nil_a ) ) ) ).
% to_list_tree_awalk
thf(fact_356_to__list__tree__single,axiom,
! [V2: list_a] :
( ( member_list_a @ V2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ? [X2: a] :
( ( V2
= ( cons_a @ X2 @ nil_a ) )
& ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% to_list_tree_single
thf(fact_357_head__del__vert,axiom,
! [U2: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ t @ U2 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_vert
thf(fact_358_head__add__vert,axiom,
! [U2: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ t @ U2 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_vert
thf(fact_359_sccs__verts__subsets,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_subsets
thf(fact_360_reachable__induce__ss,axiom,
! [S: set_a,U2: a,V2: a,T: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U2 @ V2 )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T ) @ U2 @ V2 ) ) ) ).
% reachable_induce_ss
thf(fact_361_to__list__tree__nempty,axiom,
! [V2: list_a] :
( ( member_list_a @ V2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( V2 != nil_a ) ) ).
% to_list_tree_nempty
thf(fact_362_reachable__induce__subgraphD,axiom,
! [S: set_a,U2: a,V2: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U2 @ V2 )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ U2 @ V2 ) ) ) ).
% reachable_induce_subgraphD
thf(fact_363_awalk__verts__arc2,axiom,
! [U2: a,P2: list_b,V2: a,E: b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( member_b @ E @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ).
% awalk_verts_arc2
thf(fact_364_induce__subgraph__head,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_he5236287464308401016t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= ( pre_he5236287464308401016t_unit @ G ) ) ).
% induce_subgraph_head
thf(fact_365_last__merge__is__merge,axiom,
! [Y: a] :
( ( member_a @ Y @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_366_last__merge__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z3: a] :
( ( ( reachable_a_b @ t @ X @ Z3 )
& ( Z3 != X ) )
=> ~ ( member_a @ Z3 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_367_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B3 )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_368_subset__code_I1_J,axiom,
! [Xs: list_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B3 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_369_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_370_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( member_a @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_371_subset__code_I1_J,axiom,
! [Xs: list_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B3 )
= ( ! [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
=> ( member_b @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_372_set__subset__Cons,axiom,
! [Xs: list_list_a,X: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_373_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_374_set__subset__Cons,axiom,
! [Xs: list_b,X: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ ( cons_b @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_375_awalk__to__apath__verts__subset,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% awalk_to_apath_verts_subset
thf(fact_376_not__distinct__if__head__eq__tail,axiom,
! [P2: b,U2: a,E: b,R2: a,Ps2: list_b,P22: list_b,V2: a] :
( ( ( pre_ta4931606617599662728t_unit @ t @ P2 )
= U2 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= U2 )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) @ V2 )
=> ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) ) ) ) ) ) ).
% not_distinct_if_head_eq_tail
thf(fact_377_awalk__verts__subset__if__p__sub,axiom,
! [U2: a,P1: list_b,V2: a,P22: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P1 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P22 @ V2 )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P1 ) @ ( set_b2 @ P22 ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P1 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P22 ) ) ) ) ) ) ).
% awalk_verts_subset_if_p_sub
thf(fact_378_awalk__vertex__props,axiom,
! [U2: a,P2: list_b,V2: a,P: a > $o,Q3: a > $o] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( P2 != nil_b )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( ( P @ W2 )
| ( Q3 @ W2 ) ) )
=> ( ( P @ U2 )
=> ( ( Q3 @ V2 )
=> ? [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ P2 ) )
& ( P @ ( pre_ta4931606617599662728t_unit @ t @ X2 ) )
& ( Q3 @ ( pre_he5236287464308401016t_unit @ t @ X2 ) ) ) ) ) ) ) ) ).
% awalk_vertex_props
thf(fact_379_merge__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ C2 ) )
=> ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_380_cycle__conv,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ? [U4: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ P2 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P2 ) ) )
& ( distinct_b @ P2 )
& ( P2 != nil_b ) ) ) ) ).
% cycle_conv
thf(fact_381_cas__append__if,axiom,
! [X: a,Ps2: list_b,U2: a,P2: b,V2: a] :
( ( arc_pre_cas_a_b @ t @ X @ Ps2 @ U2 )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ P2 )
= U2 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ P2 )
= V2 )
=> ( arc_pre_cas_a_b @ t @ X @ ( append_b @ Ps2 @ ( cons_b @ P2 @ nil_b ) ) @ V2 ) ) ) ) ).
% cas_append_if
thf(fact_382_tail__del__vert,axiom,
! [U2: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ t @ U2 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_vert
thf(fact_383_tail__add__vert,axiom,
! [U2: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ t @ U2 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_vert
thf(fact_384_awalk__verts__arc1,axiom,
! [E: b,P2: list_b,U2: a] :
( ( member_b @ E @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% awalk_verts_arc1
thf(fact_385_tail__and__head__eq__impl__cas,axiom,
! [X: a,P2: list_b,Y: a,G3: pre_pr7278220950009878019t_unit] :
( ( arc_pre_cas_a_b @ t @ X @ P2 @ Y )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ X2 )
= ( pre_ta4931606617599662728t_unit @ G3 @ X2 ) ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ P2 ) )
=> ( ( pre_he5236287464308401016t_unit @ t @ X2 )
= ( pre_he5236287464308401016t_unit @ G3 @ X2 ) ) )
=> ( arc_pre_cas_a_b @ G3 @ X @ P2 @ Y ) ) ) ) ).
% tail_and_head_eq_impl_cas
thf(fact_386_cas_Osimps_I2_J,axiom,
! [U2: a,E: b,Es: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ V2 )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es @ V2 ) ) ) ).
% cas.simps(2)
thf(fact_387_awalk__to__apath__subset,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) @ ( set_b2 @ P2 ) ) ) ).
% awalk_to_apath_subset
thf(fact_388_awalk__verts__arc1__app,axiom,
! [E: b,R2: a,P1: list_b,P22: list_b] : ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ ( cons_b @ E @ P22 ) ) ) ) ) ).
% awalk_verts_arc1_app
thf(fact_389_awalk__verts_Osimps_I2_J,axiom,
! [U2: a,E: b,Es: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( cons_b @ E @ Es ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es ) ) ) ).
% awalk_verts.simps(2)
thf(fact_390_cas_Oelims_I1_J,axiom,
! [X: a,Xa2: list_b,Xb: a,Y: $o] :
( ( ( arc_pre_cas_a_b @ t @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_b )
=> ( Y
= ( X != Xb ) ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( Y
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es2 @ Xb ) ) ) ) ) ) ) ).
% cas.elims(1)
thf(fact_391_cas_Oelims_I2_J,axiom,
! [X: a,Xa2: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ t @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X != Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% cas.elims(2)
thf(fact_392_cas_Oelims_I3_J,axiom,
! [X: a,Xa2: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ t @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X = Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% cas.elims(3)
thf(fact_393_distinct__tl__verts__imp__distinct,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( distinct_b @ P2 ) ) ) ).
% distinct_tl_verts_imp_distinct
thf(fact_394_cycle__altdef,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ( arc_wf_closed_w_a_b @ t @ P2 )
& ? [U4: a] : ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P2 ) ) ) ) ) ).
% cycle_altdef
thf(fact_395_cycle__def,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ? [U4: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ P2 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P2 ) ) )
& ( P2 != nil_b ) ) ) ) ).
% cycle_def
thf(fact_396_induce__subgraph__tail,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ta4931606617599662728t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= ( pre_ta4931606617599662728t_unit @ G ) ) ).
% induce_subgraph_tail
thf(fact_397_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_398_tl__append2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( tl_b @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_399_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_400_list_Ocollapse,axiom,
! [List: list_b] :
( ( List != nil_b )
=> ( ( cons_b @ ( hd_b @ List ) @ ( tl_b @ List ) )
= List ) ) ).
% list.collapse
thf(fact_401_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_402_hd__Cons__tl,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( ( cons_b @ ( hd_b @ Xs ) @ ( tl_b @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_403_subgraph__no__last__merge__chain,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( graph_8150681439568091980in_a_b @ C2 ) ) ).
% subgraph_no_last_merge_chain
thf(fact_404_wf__digraph_Oawalk__to__apath_Ocong,axiom,
arc_wf446166946845163101th_a_b = arc_wf446166946845163101th_a_b ).
% wf_digraph.awalk_to_apath.cong
thf(fact_405_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_406_list_Osel_I3_J,axiom,
! [X21: b,X22: list_b] :
( ( tl_b @ ( cons_b @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_407_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_408_list_Osel_I2_J,axiom,
( ( tl_b @ nil_b )
= nil_b ) ).
% list.sel(2)
thf(fact_409_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_410_distinct__tl,axiom,
! [Xs: list_b] :
( ( distinct_b @ Xs )
=> ( distinct_b @ ( tl_b @ Xs ) ) ) ).
% distinct_tl
thf(fact_411_pre__digraph_Ocas__simp,axiom,
! [Es: list_b,G: pre_pr2882871181989701257t_unit,U2: list_a,V2: list_a] :
( ( Es != nil_b )
=> ( ( arc_pre_cas_list_a_b @ G @ U2 @ Es @ V2 )
= ( ( ( pre_ta8437681634429857806t_unit @ G @ ( hd_b @ Es ) )
= U2 )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ ( hd_b @ Es ) ) @ ( tl_b @ Es ) @ V2 ) ) ) ) ).
% pre_digraph.cas_simp
thf(fact_412_pre__digraph_Ocas__simp,axiom,
! [Es: list_b,G: pre_pr7278220950009878019t_unit,U2: a,V2: a] :
( ( Es != nil_b )
=> ( ( arc_pre_cas_a_b @ G @ U2 @ Es @ V2 )
= ( ( ( pre_ta4931606617599662728t_unit @ G @ ( hd_b @ Es ) )
= U2 )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ ( hd_b @ Es ) ) @ ( tl_b @ Es ) @ V2 ) ) ) ) ).
% pre_digraph.cas_simp
thf(fact_413_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_414_Nil__tl,axiom,
! [Xs: list_b] :
( ( nil_b
= ( tl_b @ Xs ) )
= ( ( Xs = nil_b )
| ? [X4: b] :
( Xs
= ( cons_b @ X4 @ nil_b ) ) ) ) ).
% Nil_tl
thf(fact_415_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_416_tl__Nil,axiom,
! [Xs: list_b] :
( ( ( tl_b @ Xs )
= nil_b )
= ( ( Xs = nil_b )
| ? [X4: b] :
( Xs
= ( cons_b @ X4 @ nil_b ) ) ) ) ).
% tl_Nil
thf(fact_417_list_Oset__sel_I2_J,axiom,
! [A2: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( A2 != nil_Product_prod_a_a )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ A2 ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_418_list_Oset__sel_I2_J,axiom,
! [A2: list_set_a,X: set_a] :
( ( A2 != nil_set_a )
=> ( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ A2 ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_419_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X: a] :
( ( A2 != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a @ X @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_420_list_Oset__sel_I2_J,axiom,
! [A2: list_b,X: b] :
( ( A2 != nil_b )
=> ( ( member_b @ X @ ( set_b2 @ ( tl_b @ A2 ) ) )
=> ( member_b @ X @ ( set_b2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_421_list_Oset__sel_I2_J,axiom,
! [A2: list_list_a,X: list_a] :
( ( A2 != nil_list_a )
=> ( ( member_list_a @ X @ ( set_list_a2 @ ( tl_list_a @ A2 ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_422_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_423_tl__append__if,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( Xs = nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( tl_b @ Ys ) ) )
& ( ( Xs != nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( tl_b @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_424_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_425_list_Oexpand,axiom,
! [List: list_b,List2: list_b] :
( ( ( List = nil_b )
= ( List2 = nil_b ) )
=> ( ( ( List != nil_b )
=> ( ( List2 != nil_b )
=> ( ( ( hd_b @ List )
= ( hd_b @ List2 ) )
& ( ( tl_b @ List )
= ( tl_b @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_426_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_427_list_Oexhaust__sel,axiom,
! [List: list_b] :
( ( List != nil_b )
=> ( List
= ( cons_b @ ( hd_b @ List ) @ ( tl_b @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_428_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr3327329314391289540t_unit,U2: a,E: a,Es: list_a] :
( ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ ( cons_a @ E @ Es ) )
= ( cons_a @ ( pre_ta980714981981074249t_unit @ G @ E ) @ ( arc_pr7493981781705774525ts_a_a @ G @ ( pre_he1285395828689812537t_unit @ G @ E ) @ Es ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_429_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr3994228789931197893t_unit,U2: b,E: a,Es: list_a] :
( ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ ( cons_a @ E @ Es ) )
= ( cons_b @ ( pre_ta6449336744848955850t_unit @ G @ E ) @ ( arc_pr4706526199733098492ts_b_a @ G @ ( pre_he6754017591557694138t_unit @ G @ E ) @ Es ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_430_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr7945120425549786372t_unit,U2: b,E: b,Es: list_b] :
( ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ ( cons_b @ E @ Es ) )
= ( cons_b @ ( pre_ta1176856343612768521t_unit @ G @ E ) @ ( arc_pr4706526199733098493ts_b_b @ G @ ( pre_he1481537190321506809t_unit @ G @ E ) @ Es ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_431_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,E: b,Es: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ ( cons_b @ E @ Es ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ G @ E ) @ ( arc_pr7493981781705774526ts_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E ) @ Es ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_432_pre__digraph_Ocas_Osimps_I2_J,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a,E: b,Es: list_b,V2: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U2 @ ( cons_b @ E @ Es ) @ V2 )
= ( ( ( pre_ta8437681634429857806t_unit @ G @ E )
= U2 )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E ) @ Es @ V2 ) ) ) ).
% pre_digraph.cas.simps(2)
thf(fact_433_pre__digraph_Ocas_Osimps_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,E: b,Es: list_b,V2: a] :
( ( arc_pre_cas_a_b @ G @ U2 @ ( cons_b @ E @ Es ) @ V2 )
= ( ( ( pre_ta4931606617599662728t_unit @ G @ E )
= U2 )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E ) @ Es @ V2 ) ) ) ).
% pre_digraph.cas.simps(2)
thf(fact_434_pre__digraph_Ocas_Oelims_I1_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X: list_a,Xa2: list_b,Xb: list_a,Y: $o] :
( ( ( arc_pre_cas_list_a_b @ G @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_b )
=> ( Y
= ( X != Xb ) ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( Y
= ( ~ ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ) ) ).
% pre_digraph.cas.elims(1)
thf(fact_435_pre__digraph_Ocas_Oelims_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X: a,Xa2: list_b,Xb: a,Y: $o] :
( ( ( arc_pre_cas_a_b @ G @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_b )
=> ( Y
= ( X != Xb ) ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( Y
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ) ) ).
% pre_digraph.cas.elims(1)
thf(fact_436_pre__digraph_Ocas_Oelims_I2_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X: list_a,Xa2: list_b,Xb: list_a] :
( ( arc_pre_cas_list_a_b @ G @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X != Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ~ ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(2)
thf(fact_437_pre__digraph_Ocas_Oelims_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X: a,Xa2: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ G @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X != Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(2)
thf(fact_438_pre__digraph_Ocas_Oelims_I3_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X: list_a,Xa2: list_b,Xb: list_a] :
( ~ ( arc_pre_cas_list_a_b @ G @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X = Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(3)
thf(fact_439_pre__digraph_Ocas_Oelims_I3_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X: a,Xa2: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ G @ X @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X = Xb ) )
=> ~ ! [E2: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E2 @ Es2 ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es2 @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(3)
thf(fact_440_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,P4: list_a] :
? [U4: a] :
( ( arc_pre_awalk_a_a @ G2 @ U4 @ P4 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774525ts_a_a @ G2 @ U4 @ P4 ) ) )
& ( P4 != nil_a ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_441_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,P4: list_a] :
? [U4: b] :
( ( arc_pre_awalk_b_a @ G2 @ U4 @ P4 @ U4 )
& ( distinct_b @ ( tl_b @ ( arc_pr4706526199733098492ts_b_a @ G2 @ U4 @ P4 ) ) )
& ( P4 != nil_a ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_442_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,P4: list_b] :
? [U4: b] :
( ( arc_pre_awalk_b_b @ G2 @ U4 @ P4 @ U4 )
& ( distinct_b @ ( tl_b @ ( arc_pr4706526199733098493ts_b_b @ G2 @ U4 @ P4 ) ) )
& ( P4 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_443_pre__digraph_Ocycle__def,axiom,
( arc_pr6335352977596618620st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,P4: list_b] :
? [U4: list_a] :
( ( arc_pr6214585750886380800st_a_b @ G2 @ U4 @ P4 @ U4 )
& ( distinct_list_a @ ( tl_list_a @ ( arc_pr6350002437206376376st_a_b @ G2 @ U4 @ P4 ) ) )
& ( P4 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_444_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,P4: list_b] :
? [U4: a] :
( ( arc_pre_awalk_a_b @ G2 @ U4 @ P4 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U4 @ P4 ) ) )
& ( P4 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_445_awalkI,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 ) ) ) ) ).
% awalkI
thf(fact_446_pre__digraph_Ocas__append__if,axiom,
! [G: pre_pr2882871181989701257t_unit,X: list_a,Ps2: list_b,U2: list_a,P2: b,V2: list_a] :
( ( arc_pre_cas_list_a_b @ G @ X @ Ps2 @ U2 )
=> ( ( ( pre_ta8437681634429857806t_unit @ G @ P2 )
= U2 )
=> ( ( ( pre_he1293792728851071230t_unit @ G @ P2 )
= V2 )
=> ( arc_pre_cas_list_a_b @ G @ X @ ( append_b @ Ps2 @ ( cons_b @ P2 @ nil_b ) ) @ V2 ) ) ) ) ).
% pre_digraph.cas_append_if
thf(fact_447_pre__digraph_Ocas__append__if,axiom,
! [G: pre_pr7278220950009878019t_unit,X: a,Ps2: list_b,U2: a,P2: b,V2: a] :
( ( arc_pre_cas_a_b @ G @ X @ Ps2 @ U2 )
=> ( ( ( pre_ta4931606617599662728t_unit @ G @ P2 )
= U2 )
=> ( ( ( pre_he5236287464308401016t_unit @ G @ P2 )
= V2 )
=> ( arc_pre_cas_a_b @ G @ X @ ( append_b @ Ps2 @ ( cons_b @ P2 @ nil_b ) ) @ V2 ) ) ) ) ).
% pre_digraph.cas_append_if
thf(fact_448_awalk__verts__conv_H,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( cons_a @ U2 @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ P2 ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% awalk_verts_conv'
thf(fact_449_trail__Cons__iff,axiom,
! [U2: a,E: b,Es: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U2
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ~ ( member_b @ E @ ( set_b2 @ Es ) )
& ( arc_pre_trail_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es @ W ) ) ) ).
% trail_Cons_iff
thf(fact_450_awhd__in__verts,axiom,
! [U2: a,P2: list_b] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awhd_in_verts
thf(fact_451_awalk__def,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
= ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 ) ) ) ).
% awalk_def
thf(fact_452_arc__implies__awalk,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( arc_pre_awalk_a_b @ t @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( cons_b @ E @ nil_b ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% arc_implies_awalk
thf(fact_453_two__in__arcs__contr,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) ) ) ) ) ).
% two_in_arcs_contr
thf(fact_454_arcs__add__vert,axiom,
! [U2: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ t @ U2 ) )
= ( pre_ar1395965042833527383t_unit @ t ) ) ).
% arcs_add_vert
thf(fact_455_head__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% head_in_verts
thf(fact_456_tail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% tail_in_verts
thf(fact_457_loopfree_Ono__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ E )
!= ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% loopfree.no_loops
thf(fact_458_nomulti_Ono__multi__alt,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) )
| ( ( pre_ta4931606617599662728t_unit @ t @ E1 )
!= ( pre_ta4931606617599662728t_unit @ t @ E22 ) ) ) ) ) ) ).
% nomulti.no_multi_alt
thf(fact_459_All__arcs__in__path,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [P5: list_b,U5: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U5 @ P5 @ V )
& ( member_b @ E @ ( set_b2 @ P5 ) ) ) ) ).
% All_arcs_in_path
thf(fact_460_inner__verts__def,axiom,
! [P2: list_b] :
( ( pre_inner_verts_a_b @ t @ P2 )
= ( tl_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) ).
% inner_verts_def
thf(fact_461_awalk__Cons__iff,axiom,
! [U2: a,E: b,Es: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U2
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ( arc_pre_awalk_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es @ W ) ) ) ).
% awalk_Cons_iff
thf(fact_462_cas__simp,axiom,
! [Es: list_b,U2: a,V2: a] :
( ( Es != nil_b )
=> ( ( arc_pre_cas_a_b @ t @ U2 @ Es @ V2 )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ Es ) )
= U2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ ( hd_b @ Es ) ) @ ( tl_b @ Es ) @ V2 ) ) ) ) ).
% cas_simp
thf(fact_463_awalk__verts__in__verts,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_a @ V2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
=> ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ).
% awalk_verts_in_verts
thf(fact_464_map__eq__conv,axiom,
! [F: b > a,Xs: list_b,G4: b > a] :
( ( ( map_b_a @ F @ Xs )
= ( map_b_a @ G4 @ Xs ) )
= ( ! [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
=> ( ( F @ X4 )
= ( G4 @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_465_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_466_map__append,axiom,
! [F: a > b,Xs: list_a,Ys: list_a] :
( ( map_a_b @ F @ ( append_a @ Xs @ Ys ) )
= ( append_b @ ( map_a_b @ F @ Xs ) @ ( map_a_b @ F @ Ys ) ) ) ).
% map_append
thf(fact_467_map__append,axiom,
! [F: b > b,Xs: list_b,Ys: list_b] :
( ( map_b_b @ F @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( map_b_b @ F @ Xs ) @ ( map_b_b @ F @ Ys ) ) ) ).
% map_append
thf(fact_468_map__append,axiom,
! [F: b > a,Xs: list_b,Ys: list_b] :
( ( map_b_a @ F @ ( append_b @ Xs @ Ys ) )
= ( append_a @ ( map_b_a @ F @ Xs ) @ ( map_b_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_469_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_470_map__is__Nil__conv,axiom,
! [F: a > b,Xs: list_a] :
( ( ( map_a_b @ F @ Xs )
= nil_b )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_471_map__is__Nil__conv,axiom,
! [F: b > b,Xs: list_b] :
( ( ( map_b_b @ F @ Xs )
= nil_b )
= ( Xs = nil_b ) ) ).
% map_is_Nil_conv
thf(fact_472_map__is__Nil__conv,axiom,
! [F: b > a,Xs: list_b] :
( ( ( map_b_a @ F @ Xs )
= nil_a )
= ( Xs = nil_b ) ) ).
% map_is_Nil_conv
thf(fact_473_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_474_Nil__is__map__conv,axiom,
! [F: a > b,Xs: list_a] :
( ( nil_b
= ( map_a_b @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_475_Nil__is__map__conv,axiom,
! [F: b > b,Xs: list_b] :
( ( nil_b
= ( map_b_b @ F @ Xs ) )
= ( Xs = nil_b ) ) ).
% Nil_is_map_conv
thf(fact_476_Nil__is__map__conv,axiom,
! [F: b > a,Xs: list_b] :
( ( nil_a
= ( map_b_a @ F @ Xs ) )
= ( Xs = nil_b ) ) ).
% Nil_is_map_conv
thf(fact_477_list_Omap__disc__iff,axiom,
! [F: a > a,A2: list_a] :
( ( ( map_a_a @ F @ A2 )
= nil_a )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_478_list_Omap__disc__iff,axiom,
! [F: a > b,A2: list_a] :
( ( ( map_a_b @ F @ A2 )
= nil_b )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_479_list_Omap__disc__iff,axiom,
! [F: b > b,A2: list_b] :
( ( ( map_b_b @ F @ A2 )
= nil_b )
= ( A2 = nil_b ) ) ).
% list.map_disc_iff
thf(fact_480_list_Omap__disc__iff,axiom,
! [F: b > a,A2: list_b] :
( ( ( map_b_a @ F @ A2 )
= nil_a )
= ( A2 = nil_b ) ) ).
% list.map_disc_iff
thf(fact_481_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_482_map__eq__Cons__conv,axiom,
! [F: b > a,Xs: list_b,Y: a,Ys: list_a] :
( ( ( map_b_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: b,Zs2: list_b] :
( ( Xs
= ( cons_b @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_b_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_483_map__eq__Cons__conv,axiom,
! [F: a > b,Xs: list_a,Y: b,Ys: list_b] :
( ( ( map_a_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_b @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_484_map__eq__Cons__conv,axiom,
! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
( ( ( map_b_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
= ( ? [Z4: b,Zs2: list_b] :
( ( Xs
= ( cons_b @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_b_b @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_485_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_486_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: b > a,Ys: list_b] :
( ( ( cons_a @ X @ Xs )
= ( map_b_a @ F @ Ys ) )
= ( ? [Z4: b,Zs2: list_b] :
( ( Ys
= ( cons_b @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_b_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_487_Cons__eq__map__conv,axiom,
! [X: b,Xs: list_b,F: a > b,Ys: list_a] :
( ( ( cons_b @ X @ Xs )
= ( map_a_b @ F @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_a_b @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_488_Cons__eq__map__conv,axiom,
! [X: b,Xs: list_b,F: b > b,Ys: list_b] :
( ( ( cons_b @ X @ Xs )
= ( map_b_b @ F @ Ys ) )
= ( ? [Z4: b,Zs2: list_b] :
( ( Ys
= ( cons_b @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_b_b @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_489_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z5: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_a_a @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_490_map__eq__Cons__D,axiom,
! [F: b > a,Xs: list_b,Y: a,Ys: list_a] :
( ( ( map_b_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z5: b,Zs3: list_b] :
( ( Xs
= ( cons_b @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_b_a @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_491_map__eq__Cons__D,axiom,
! [F: a > b,Xs: list_a,Y: b,Ys: list_b] :
( ( ( map_a_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
=> ? [Z5: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_a_b @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_492_map__eq__Cons__D,axiom,
! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
( ( ( map_b_b @ F @ Xs )
= ( cons_b @ Y @ Ys ) )
=> ? [Z5: b,Zs3: list_b] :
( ( Xs
= ( cons_b @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_b_b @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_493_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z5: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_a_a @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_494_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: b > a,Ys: list_b] :
( ( ( cons_a @ X @ Xs )
= ( map_b_a @ F @ Ys ) )
=> ? [Z5: b,Zs3: list_b] :
( ( Ys
= ( cons_b @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_b_a @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_495_Cons__eq__map__D,axiom,
! [X: b,Xs: list_b,F: a > b,Ys: list_a] :
( ( ( cons_b @ X @ Xs )
= ( map_a_b @ F @ Ys ) )
=> ? [Z5: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_a_b @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_496_Cons__eq__map__D,axiom,
! [X: b,Xs: list_b,F: b > b,Ys: list_b] :
( ( ( cons_b @ X @ Xs )
= ( map_b_b @ F @ Ys ) )
=> ? [Z5: b,Zs3: list_b] :
( ( Ys
= ( cons_b @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_b_b @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_497_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_498_list_Osimps_I9_J,axiom,
! [F: a > b,X21: a,X22: list_a] :
( ( map_a_b @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_b @ ( F @ X21 ) @ ( map_a_b @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_499_list_Osimps_I9_J,axiom,
! [F: b > a,X21: b,X22: list_b] :
( ( map_b_a @ F @ ( cons_b @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_b_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_500_list_Osimps_I9_J,axiom,
! [F: b > b,X21: b,X22: list_b] :
( ( map_b_b @ F @ ( cons_b @ X21 @ X22 ) )
= ( cons_b @ ( F @ X21 ) @ ( map_b_b @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_501_ex__map__conv,axiom,
! [Ys: list_a,F: b > a] :
( ( ? [Xs4: list_b] :
( Ys
= ( map_b_a @ F @ Xs4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Ys ) )
=> ? [Y3: b] :
( X4
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_502_map__cong,axiom,
! [Xs: list_b,Ys: list_b,F: b > a,G4: b > a] :
( ( Xs = Ys )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ Ys ) )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( map_b_a @ F @ Xs )
= ( map_b_a @ G4 @ Ys ) ) ) ) ).
% map_cong
thf(fact_503_map__idI,axiom,
! [Xs: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Pr7904243085458786820od_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_504_map__idI,axiom,
! [Xs: list_set_a,F: set_a > set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_set_a_set_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_505_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_506_map__idI,axiom,
! [Xs: list_b,F: b > b] :
( ! [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_b_b @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_507_map__idI,axiom,
! [Xs: list_list_a,F: list_a > list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_list_a_list_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_508_map__ext,axiom,
! [Xs: list_b,F: b > a,G4: b > a] :
( ! [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( map_b_a @ F @ Xs )
= ( map_b_a @ G4 @ Xs ) ) ) ).
% map_ext
thf(fact_509_list_Omap__ident__strong,axiom,
! [T2: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [Z5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z5 @ ( set_Product_prod_a_a2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_Pr7904243085458786820od_a_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_510_list_Omap__ident__strong,axiom,
! [T2: list_set_a,F: set_a > set_a] :
( ! [Z5: set_a] :
( ( member_set_a @ Z5 @ ( set_set_a2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_set_a_set_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_511_list_Omap__ident__strong,axiom,
! [T2: list_a,F: a > a] :
( ! [Z5: a] :
( ( member_a @ Z5 @ ( set_a2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_a_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_512_list_Omap__ident__strong,axiom,
! [T2: list_b,F: b > b] :
( ! [Z5: b] :
( ( member_b @ Z5 @ ( set_b2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_b_b @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_513_list_Omap__ident__strong,axiom,
! [T2: list_list_a,F: list_a > list_a] :
( ! [Z5: list_a] :
( ( member_list_a @ Z5 @ ( set_list_a2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_list_a_list_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_514_list_Oinj__map__strong,axiom,
! [X: list_b,Xa2: list_b,F: b > a,Fa: b > a] :
( ! [Z5: b,Za: b] :
( ( member_b @ Z5 @ ( set_b2 @ X ) )
=> ( ( member_b @ Za @ ( set_b2 @ Xa2 ) )
=> ( ( ( F @ Z5 )
= ( Fa @ Za ) )
=> ( Z5 = Za ) ) ) )
=> ( ( ( map_b_a @ F @ X )
= ( map_b_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_515_list_Omap__cong0,axiom,
! [X: list_b,F: b > a,G4: b > a] :
( ! [Z5: b] :
( ( member_b @ Z5 @ ( set_b2 @ X ) )
=> ( ( F @ Z5 )
= ( G4 @ Z5 ) ) )
=> ( ( map_b_a @ F @ X )
= ( map_b_a @ G4 @ X ) ) ) ).
% list.map_cong0
thf(fact_516_list_Omap__cong,axiom,
! [X: list_b,Ya: list_b,F: b > a,G4: b > a] :
( ( X = Ya )
=> ( ! [Z5: b] :
( ( member_b @ Z5 @ ( set_b2 @ Ya ) )
=> ( ( F @ Z5 )
= ( G4 @ Z5 ) ) )
=> ( ( map_b_a @ F @ X )
= ( map_b_a @ G4 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_517_map__eq__append__conv,axiom,
! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_518_map__eq__append__conv,axiom,
! [F: a > b,Xs: list_a,Ys: list_b,Zs: list_b] :
( ( ( map_a_b @ F @ Xs )
= ( append_b @ Ys @ Zs ) )
= ( ? [Us: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us @ Vs2 ) )
& ( Ys
= ( map_a_b @ F @ Us ) )
& ( Zs
= ( map_a_b @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_519_map__eq__append__conv,axiom,
! [F: b > b,Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( map_b_b @ F @ Xs )
= ( append_b @ Ys @ Zs ) )
= ( ? [Us: list_b,Vs2: list_b] :
( ( Xs
= ( append_b @ Us @ Vs2 ) )
& ( Ys
= ( map_b_b @ F @ Us ) )
& ( Zs
= ( map_b_b @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_520_map__eq__append__conv,axiom,
! [F: b > a,Xs: list_b,Ys: list_a,Zs: list_a] :
( ( ( map_b_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us: list_b,Vs2: list_b] :
( ( Xs
= ( append_b @ Us @ Vs2 ) )
& ( Ys
= ( map_b_a @ F @ Us ) )
& ( Zs
= ( map_b_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_521_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_a_a @ F @ Xs ) )
= ( ? [Us: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_522_append__eq__map__conv,axiom,
! [Ys: list_b,Zs: list_b,F: a > b,Xs: list_a] :
( ( ( append_b @ Ys @ Zs )
= ( map_a_b @ F @ Xs ) )
= ( ? [Us: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us @ Vs2 ) )
& ( Ys
= ( map_a_b @ F @ Us ) )
& ( Zs
= ( map_a_b @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_523_append__eq__map__conv,axiom,
! [Ys: list_b,Zs: list_b,F: b > b,Xs: list_b] :
( ( ( append_b @ Ys @ Zs )
= ( map_b_b @ F @ Xs ) )
= ( ? [Us: list_b,Vs2: list_b] :
( ( Xs
= ( append_b @ Us @ Vs2 ) )
& ( Ys
= ( map_b_b @ F @ Us ) )
& ( Zs
= ( map_b_b @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_524_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: b > a,Xs: list_b] :
( ( ( append_a @ Ys @ Zs )
= ( map_b_a @ F @ Xs ) )
= ( ? [Us: list_b,Vs2: list_b] :
( ( Xs
= ( append_b @ Us @ Vs2 ) )
& ( Ys
= ( map_b_a @ F @ Us ) )
& ( Zs
= ( map_b_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_525_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_526_list_Osimps_I8_J,axiom,
! [F: a > b] :
( ( map_a_b @ F @ nil_a )
= nil_b ) ).
% list.simps(8)
thf(fact_527_list_Osimps_I8_J,axiom,
! [F: b > b] :
( ( map_b_b @ F @ nil_b )
= nil_b ) ).
% list.simps(8)
thf(fact_528_list_Osimps_I8_J,axiom,
! [F: b > a] :
( ( map_b_a @ F @ nil_b )
= nil_a ) ).
% list.simps(8)
thf(fact_529_rev__map,axiom,
! [F: a > a,Xs: list_a] :
( ( rev_a @ ( map_a_a @ F @ Xs ) )
= ( map_a_a @ F @ ( rev_a @ Xs ) ) ) ).
% rev_map
thf(fact_530_rev__map,axiom,
! [F: b > a,Xs: list_b] :
( ( rev_a @ ( map_b_a @ F @ Xs ) )
= ( map_b_a @ F @ ( rev_b @ Xs ) ) ) ).
% rev_map
thf(fact_531_map__tl,axiom,
! [F: a > a,Xs: list_a] :
( ( map_a_a @ F @ ( tl_a @ Xs ) )
= ( tl_a @ ( map_a_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_532_map__tl,axiom,
! [F: a > b,Xs: list_a] :
( ( map_a_b @ F @ ( tl_a @ Xs ) )
= ( tl_b @ ( map_a_b @ F @ Xs ) ) ) ).
% map_tl
thf(fact_533_map__tl,axiom,
! [F: b > b,Xs: list_b] :
( ( map_b_b @ F @ ( tl_b @ Xs ) )
= ( tl_b @ ( map_b_b @ F @ Xs ) ) ) ).
% map_tl
thf(fact_534_map__tl,axiom,
! [F: b > a,Xs: list_b] :
( ( map_b_a @ F @ ( tl_b @ Xs ) )
= ( tl_a @ ( map_b_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_535_list_Omap__sel_I1_J,axiom,
! [A2: list_a,F: a > a] :
( ( A2 != nil_a )
=> ( ( hd_a @ ( map_a_a @ F @ A2 ) )
= ( F @ ( hd_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_536_list_Omap__sel_I1_J,axiom,
! [A2: list_a,F: a > b] :
( ( A2 != nil_a )
=> ( ( hd_b @ ( map_a_b @ F @ A2 ) )
= ( F @ ( hd_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_537_list_Omap__sel_I1_J,axiom,
! [A2: list_b,F: b > b] :
( ( A2 != nil_b )
=> ( ( hd_b @ ( map_b_b @ F @ A2 ) )
= ( F @ ( hd_b @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_538_list_Omap__sel_I1_J,axiom,
! [A2: list_b,F: b > a] :
( ( A2 != nil_b )
=> ( ( hd_a @ ( map_b_a @ F @ A2 ) )
= ( F @ ( hd_b @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_539_hd__map,axiom,
! [Xs: list_a,F: a > a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( map_a_a @ F @ Xs ) )
= ( F @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_540_hd__map,axiom,
! [Xs: list_a,F: a > b] :
( ( Xs != nil_a )
=> ( ( hd_b @ ( map_a_b @ F @ Xs ) )
= ( F @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_541_hd__map,axiom,
! [Xs: list_b,F: b > b] :
( ( Xs != nil_b )
=> ( ( hd_b @ ( map_b_b @ F @ Xs ) )
= ( F @ ( hd_b @ Xs ) ) ) ) ).
% hd_map
thf(fact_542_hd__map,axiom,
! [Xs: list_b,F: b > a] :
( ( Xs != nil_b )
=> ( ( hd_a @ ( map_b_a @ F @ Xs ) )
= ( F @ ( hd_b @ Xs ) ) ) ) ).
% hd_map
thf(fact_543_list_Omap__sel_I2_J,axiom,
! [A2: list_a,F: a > a] :
( ( A2 != nil_a )
=> ( ( tl_a @ ( map_a_a @ F @ A2 ) )
= ( map_a_a @ F @ ( tl_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_544_list_Omap__sel_I2_J,axiom,
! [A2: list_a,F: a > b] :
( ( A2 != nil_a )
=> ( ( tl_b @ ( map_a_b @ F @ A2 ) )
= ( map_a_b @ F @ ( tl_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_545_list_Omap__sel_I2_J,axiom,
! [A2: list_b,F: b > b] :
( ( A2 != nil_b )
=> ( ( tl_b @ ( map_b_b @ F @ A2 ) )
= ( map_b_b @ F @ ( tl_b @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_546_list_Omap__sel_I2_J,axiom,
! [A2: list_b,F: b > a] :
( ( A2 != nil_b )
=> ( ( tl_a @ ( map_b_a @ F @ A2 ) )
= ( map_b_a @ F @ ( tl_b @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_547_directed__tree_Oto__list__tree_Ocong,axiom,
direct3773525127397338803ee_a_b = direct3773525127397338803ee_a_b ).
% directed_tree.to_list_tree.cong
thf(fact_548_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,U4: b,P4: list_a,V5: b] :
( ( member_b @ U4 @ ( pre_ve2160112157898065374t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( pre_ar2913695170082820505t_unit @ G2 ) )
& ( arc_pre_cas_b_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_549_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,U4: a,P4: list_a,V5: a] :
( ( member_a @ U4 @ ( pre_ve5914862431884959581t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( pre_ar6668445444069714712t_unit @ G2 ) )
& ( arc_pre_cas_a_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_550_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,U4: b,P4: list_b,V5: b] :
( ( member_b @ U4 @ ( pre_ve6111003793516653853t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P4 ) @ ( pre_ar6864586805701408984t_unit @ G2 ) )
& ( arc_pre_cas_b_b @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_551_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P4: list_b,V5: a] :
( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P4 ) @ ( pre_ar1395965042833527383t_unit @ G2 ) )
& ( arc_pre_cas_a_b @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_552_pre__digraph_Oawalk__def,axiom,
( arc_pr368411507155669706list_a
= ( ^ [G2: pre_pr7651200976661991615t_unit,U4: b,P4: list_list_a,V5: b] :
( ( member_b @ U4 @ ( pre_ve7998398825243057368t_unit @ G2 ) )
& ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( pre_ar405773122724138771t_unit @ G2 ) )
& ( arc_pre_cas_b_list_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_553_pre__digraph_Oawalk__def,axiom,
( arc_pr8103821506715646987list_a
= ( ^ [G2: pre_pr3711252390037155390t_unit,U4: a,P4: list_list_a,V5: a] :
( ( member_a @ U4 @ ( pre_ve3018455677094554327t_unit @ G2 ) )
& ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( pre_ar4649202011430411538t_unit @ G2 ) )
& ( arc_pre_cas_a_list_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_554_pre__digraph_Oawalk__def,axiom,
( arc_pr6214585750886380799st_a_a
= ( ^ [G2: pre_pr8155351583225888586t_unit,U4: list_a,P4: list_a,V5: list_a] :
( ( member_list_a @ U4 @ ( pre_ve7102540449451629283t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( pre_ar8733286783787486494t_unit @ G2 ) )
& ( arc_pre_cas_list_a_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_555_pre__digraph_Oawalk__def,axiom,
( arc_pr441381926571271589et_a_a
= ( ^ [G2: pre_pr3647964229410195492t_unit,U4: set_a,P4: list_a,V5: set_a] :
( ( member_set_a @ U4 @ ( pre_ve2608818176351713469t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( pre_ar4979499625094109304t_unit @ G2 ) )
& ( arc_pre_cas_set_a_a @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_556_pre__digraph_Oawalk__def,axiom,
( arc_pr441381926571271590et_a_b
= ( ^ [G2: pre_pr7598855865028783971t_unit,U4: set_a,P4: list_b,V5: set_a] :
( ( member_set_a @ U4 @ ( pre_ve6559709811970301948t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P4 ) @ ( pre_ar8930391260712697783t_unit @ G2 ) )
& ( arc_pre_cas_set_a_b @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_557_pre__digraph_Oawalk__def,axiom,
( arc_pr6214585750886380800st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P4: list_b,V5: list_a] :
( ( member_list_a @ U4 @ ( pre_ve1830060048215441954t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P4 ) @ ( pre_ar3460806382551299165t_unit @ G2 ) )
& ( arc_pre_cas_list_a_b @ G2 @ U4 @ P4 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_558_pre__digraph_Oawalk__verts__conv_H,axiom,
! [G: pre_pr3327329314391289540t_unit,U2: a,P2: list_a,V2: a] :
( ( arc_pre_cas_a_a @ G @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_a )
=> ( ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ P2 )
= ( cons_a @ U2 @ nil_a ) ) )
& ( ( P2 != nil_a )
=> ( ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ P2 )
= ( cons_a @ ( pre_ta980714981981074249t_unit @ G @ ( hd_a @ P2 ) ) @ ( map_a_a @ ( pre_he1285395828689812537t_unit @ G ) @ P2 ) ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv'
thf(fact_559_pre__digraph_Oawalk__verts__conv_H,axiom,
! [G: pre_pr3994228789931197893t_unit,U2: b,P2: list_a,V2: b] :
( ( arc_pre_cas_b_a @ G @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_a )
=> ( ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ P2 )
= ( cons_b @ U2 @ nil_b ) ) )
& ( ( P2 != nil_a )
=> ( ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ P2 )
= ( cons_b @ ( pre_ta6449336744848955850t_unit @ G @ ( hd_a @ P2 ) ) @ ( map_a_b @ ( pre_he6754017591557694138t_unit @ G ) @ P2 ) ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv'
thf(fact_560_pre__digraph_Oawalk__verts__conv_H,axiom,
! [G: pre_pr7945120425549786372t_unit,U2: b,P2: list_b,V2: b] :
( ( arc_pre_cas_b_b @ G @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ P2 )
= ( cons_b @ U2 @ nil_b ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ P2 )
= ( cons_b @ ( pre_ta1176856343612768521t_unit @ G @ ( hd_b @ P2 ) ) @ ( map_b_b @ ( pre_he1481537190321506809t_unit @ G ) @ P2 ) ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv'
thf(fact_561_pre__digraph_Oawalk__verts__conv_H,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a,P2: list_b,V2: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr6350002437206376376st_a_b @ G @ U2 @ P2 )
= ( cons_list_a @ U2 @ nil_list_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr6350002437206376376st_a_b @ G @ U2 @ P2 )
= ( cons_list_a @ ( pre_ta8437681634429857806t_unit @ G @ ( hd_b @ P2 ) ) @ ( map_b_list_a @ ( pre_he1293792728851071230t_unit @ G ) @ P2 ) ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv'
thf(fact_562_pre__digraph_Oawalk__verts__conv_H,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ G @ U2 @ P2 @ V2 )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 )
= ( cons_a @ U2 @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ G @ ( hd_b @ P2 ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ G ) @ P2 ) ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv'
thf(fact_563_euler__trail__def,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( pre_euler_trail_a_b @ t @ U2 @ P2 @ V2 )
= ( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ V2 )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% euler_trail_def
thf(fact_564_awalk__conv,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
= ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= U2 )
& ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= V2 )
& ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 ) ) ) ).
% awalk_conv
thf(fact_565_awalkE_H,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= U2 )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= V2 )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ~ ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ) ) ) ) ).
% awalkE'
thf(fact_566_awalkE,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= U2 )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
!= V2 ) ) ) ) ) ) ).
% awalkE
thf(fact_567_set__awalk__verts__not__Nil__cas,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil_cas
thf(fact_568_set__awalk__verts__not__Nil,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil
thf(fact_569_awlast__append,axiom,
! [U2: a,P2: list_b,Q: list_b] :
( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) ) )
= ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q ) ) ) ).
% awlast_append
thf(fact_570_awlast__if__cas,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= V2 ) ) ).
% awlast_if_cas
thf(fact_571_fwd__arcs__conc__nlast__elem,axiom,
! [Xs: list_a,Y: a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs )
=> ( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ( ( Y
!= ( last_a @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) ) ) ) ) ).
% fwd_arcs_conc_nlast_elem
thf(fact_572_set__awalk__verts__append,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ V2 @ Q @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V2 @ Q ) ) ) ) ) ) ).
% set_awalk_verts_append
thf(fact_573_set__awalk__verts__append__cas,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b,W: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_cas_a_b @ t @ V2 @ Q @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V2 @ Q ) ) ) ) ) ) ).
% set_awalk_verts_append_cas
thf(fact_574_awlast__in__verts,axiom,
! [U2: a,P2: list_b] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awlast_in_verts
thf(fact_575_awalk__verts__append,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q ) ) ) ) ) ).
% awalk_verts_append
thf(fact_576_awalk__verts__append__cas,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q ) ) ) ) ) ).
% awalk_verts_append_cas
thf(fact_577_set__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( set_b2 @ ( append_b @ Xs @ Ys ) )
= ( sup_sup_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) ) ) ).
% set_append
thf(fact_578_set__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( set_list_a2 @ ( append_list_a @ Xs @ Ys ) )
= ( sup_sup_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) ) ) ).
% set_append
thf(fact_579_set__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( append_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_580_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_581_last__appendL,axiom,
! [Ys: list_b,Xs: list_b] :
( ( Ys = nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Xs ) ) ) ).
% last_appendL
thf(fact_582_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_583_last__appendR,axiom,
! [Ys: list_b,Xs: list_b] :
( ( Ys != nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Ys ) ) ) ).
% last_appendR
thf(fact_584_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_585_last__snoc,axiom,
! [Xs: list_b,X: b] :
( ( last_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
= X ) ).
% last_snoc
thf(fact_586_awalk__append__iff,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
& ( arc_pre_awalk_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% awalk_append_iff
thf(fact_587_cas__append__iff,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
& ( arc_pre_cas_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% cas_append_iff
thf(fact_588_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_589_last_Osimps,axiom,
! [Xs: list_b,X: b] :
( ( ( Xs = nil_b )
=> ( ( last_b @ ( cons_b @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_b )
=> ( ( last_b @ ( cons_b @ X @ Xs ) )
= ( last_b @ Xs ) ) ) ) ).
% last.simps
thf(fact_590_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_591_last__ConsL,axiom,
! [Xs: list_b,X: b] :
( ( Xs = nil_b )
=> ( ( last_b @ ( cons_b @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_592_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_593_last__ConsR,axiom,
! [Xs: list_b,X: b] :
( ( Xs != nil_b )
=> ( ( last_b @ ( cons_b @ X @ Xs ) )
= ( last_b @ Xs ) ) ) ).
% last_ConsR
thf(fact_594_last__in__set,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( As != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ As ) @ ( set_Product_prod_a_a2 @ As ) ) ) ).
% last_in_set
thf(fact_595_last__in__set,axiom,
! [As: list_set_a] :
( ( As != nil_set_a )
=> ( member_set_a @ ( last_set_a @ As ) @ ( set_set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_596_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_597_last__in__set,axiom,
! [As: list_b] :
( ( As != nil_b )
=> ( member_b @ ( last_b @ As ) @ ( set_b2 @ As ) ) ) ).
% last_in_set
thf(fact_598_last__in__set,axiom,
! [As: list_list_a] :
( ( As != nil_list_a )
=> ( member_list_a @ ( last_list_a @ As ) @ ( set_list_a2 @ As ) ) ) ).
% last_in_set
thf(fact_599_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_600_last__append,axiom,
! [Ys: list_b,Xs: list_b] :
( ( ( Ys = nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Xs ) ) )
& ( ( Ys != nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Ys ) ) ) ) ).
% last_append
thf(fact_601_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_602_longest__common__suffix,axiom,
! [Xs: list_b,Ys: list_b] :
? [Ss: list_b,Xs3: list_b,Ys5: list_b] :
( ( Xs
= ( append_b @ Xs3 @ Ss ) )
& ( Ys
= ( append_b @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_b )
| ( Ys5 = nil_b )
| ( ( last_b @ Xs3 )
!= ( last_b @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_603_last__map,axiom,
! [Xs: list_a,F: a > a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( map_a_a @ F @ Xs ) )
= ( F @ ( last_a @ Xs ) ) ) ) ).
% last_map
thf(fact_604_last__map,axiom,
! [Xs: list_a,F: a > b] :
( ( Xs != nil_a )
=> ( ( last_b @ ( map_a_b @ F @ Xs ) )
= ( F @ ( last_a @ Xs ) ) ) ) ).
% last_map
thf(fact_605_last__map,axiom,
! [Xs: list_b,F: b > b] :
( ( Xs != nil_b )
=> ( ( last_b @ ( map_b_b @ F @ Xs ) )
= ( F @ ( last_b @ Xs ) ) ) ) ).
% last_map
thf(fact_606_last__map,axiom,
! [Xs: list_b,F: b > a] :
( ( Xs != nil_b )
=> ( ( last_a @ ( map_b_a @ F @ Xs ) )
= ( F @ ( last_b @ Xs ) ) ) ) ).
% last_map
thf(fact_607_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_608_hd__Nil__eq__last,axiom,
( ( hd_b @ nil_b )
= ( last_b @ nil_b ) ) ).
% hd_Nil_eq_last
thf(fact_609_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_610_last__rev,axiom,
! [Xs: list_b] :
( ( last_b @ ( rev_b @ Xs ) )
= ( hd_b @ Xs ) ) ).
% last_rev
thf(fact_611_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_612_hd__rev,axiom,
! [Xs: list_b] :
( ( hd_b @ ( rev_b @ Xs ) )
= ( last_b @ Xs ) ) ).
% hd_rev
thf(fact_613_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_614_last__tl,axiom,
! [Xs: list_b] :
( ( ( Xs = nil_b )
| ( ( tl_b @ Xs )
!= nil_b ) )
=> ( ( last_b @ ( tl_b @ Xs ) )
= ( last_b @ Xs ) ) ) ).
% last_tl
thf(fact_615_split__last__eq,axiom,
! [As: list_a,Y: a,Bs: list_a,Xs: list_a] :
( ( ( append_a @ As @ ( cons_a @ Y @ Bs ) )
= Xs )
=> ( ( Bs != nil_a )
=> ( ( last_a @ Bs )
= ( last_a @ Xs ) ) ) ) ).
% split_last_eq
thf(fact_616_split__last__eq,axiom,
! [As: list_b,Y: b,Bs: list_b,Xs: list_b] :
( ( ( append_b @ As @ ( cons_b @ Y @ Bs ) )
= Xs )
=> ( ( Bs != nil_b )
=> ( ( last_b @ Bs )
= ( last_b @ Xs ) ) ) ) ).
% split_last_eq
thf(fact_617_pre__digraph_Ocas__append__iff,axiom,
! [G: pre_pr3327329314391289540t_unit,U2: a,P2: list_a,Q: list_a,V2: a] :
( ( arc_pre_cas_a_a @ G @ U2 @ ( append_a @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_a_a @ G @ U2 @ P2 @ ( last_a @ ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ P2 ) ) )
& ( arc_pre_cas_a_a @ G @ ( last_a @ ( arc_pr7493981781705774525ts_a_a @ G @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% pre_digraph.cas_append_iff
thf(fact_618_pre__digraph_Ocas__append__iff,axiom,
! [G: pre_pr3994228789931197893t_unit,U2: b,P2: list_a,Q: list_a,V2: b] :
( ( arc_pre_cas_b_a @ G @ U2 @ ( append_a @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_b_a @ G @ U2 @ P2 @ ( last_b @ ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ P2 ) ) )
& ( arc_pre_cas_b_a @ G @ ( last_b @ ( arc_pr4706526199733098492ts_b_a @ G @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% pre_digraph.cas_append_iff
thf(fact_619_pre__digraph_Ocas__append__iff,axiom,
! [G: pre_pr7945120425549786372t_unit,U2: b,P2: list_b,Q: list_b,V2: b] :
( ( arc_pre_cas_b_b @ G @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_b_b @ G @ U2 @ P2 @ ( last_b @ ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ P2 ) ) )
& ( arc_pre_cas_b_b @ G @ ( last_b @ ( arc_pr4706526199733098493ts_b_b @ G @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% pre_digraph.cas_append_iff
thf(fact_620_pre__digraph_Ocas__append__iff,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a,P2: list_b,Q: list_b,V2: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_list_a_b @ G @ U2 @ P2 @ ( last_list_a @ ( arc_pr6350002437206376376st_a_b @ G @ U2 @ P2 ) ) )
& ( arc_pre_cas_list_a_b @ G @ ( last_list_a @ ( arc_pr6350002437206376376st_a_b @ G @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% pre_digraph.cas_append_iff
thf(fact_621_pre__digraph_Ocas__append__iff,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_cas_a_b @ G @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_cas_a_b @ G @ U2 @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 ) ) )
& ( arc_pre_cas_a_b @ G @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ G @ U2 @ P2 ) ) @ Q @ V2 ) ) ) ).
% pre_digraph.cas_append_iff
thf(fact_622_split__list__last__sep,axiom,
! [Y: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( Y
!= ( last_P8790725268278465478od_a_a @ Xs ) )
=> ? [As2: list_P1396940483166286381od_a_a,Bs2: list_P1396940483166286381od_a_a] :
( ( append5335208819046833346od_a_a @ As2 @ ( cons_P7316939126706565853od_a_a @ Y @ ( append5335208819046833346od_a_a @ Bs2 @ ( cons_P7316939126706565853od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ nil_Product_prod_a_a ) ) ) )
= Xs ) ) ) ).
% split_list_last_sep
thf(fact_623_split__list__last__sep,axiom,
! [Y: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ Xs ) )
=> ( ( Y
!= ( last_set_a @ Xs ) )
=> ? [As2: list_set_a,Bs2: list_set_a] :
( ( append_set_a @ As2 @ ( cons_set_a @ Y @ ( append_set_a @ Bs2 @ ( cons_set_a @ ( last_set_a @ Xs ) @ nil_set_a ) ) ) )
= Xs ) ) ) ).
% split_list_last_sep
thf(fact_624_split__list__last__sep,axiom,
! [Y: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ Xs ) )
=> ( ( Y
!= ( last_list_a @ Xs ) )
=> ? [As2: list_list_a,Bs2: list_list_a] :
( ( append_list_a @ As2 @ ( cons_list_a @ Y @ ( append_list_a @ Bs2 @ ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) ) ) )
= Xs ) ) ) ).
% split_list_last_sep
thf(fact_625_split__list__last__sep,axiom,
! [Y: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ( ( Y
!= ( last_a @ Xs ) )
=> ? [As2: list_a,Bs2: list_a] :
( ( append_a @ As2 @ ( cons_a @ Y @ ( append_a @ Bs2 @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) )
= Xs ) ) ) ).
% split_list_last_sep
thf(fact_626_split__list__last__sep,axiom,
! [Y: b,Xs: list_b] :
( ( member_b @ Y @ ( set_b2 @ Xs ) )
=> ( ( Y
!= ( last_b @ Xs ) )
=> ? [As2: list_b,Bs2: list_b] :
( ( append_b @ As2 @ ( cons_b @ Y @ ( append_b @ Bs2 @ ( cons_b @ ( last_b @ Xs ) @ nil_b ) ) ) )
= Xs ) ) ) ).
% split_list_last_sep
thf(fact_627_split__list__not__last,axiom,
! [Y: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( Y
!= ( last_P8790725268278465478od_a_a @ Xs ) )
=> ? [As2: list_P1396940483166286381od_a_a,Bs2: list_P1396940483166286381od_a_a] :
( ( ( append5335208819046833346od_a_a @ As2 @ ( cons_P7316939126706565853od_a_a @ Y @ Bs2 ) )
= Xs )
& ( Bs2 != nil_Product_prod_a_a ) ) ) ) ).
% split_list_not_last
thf(fact_628_split__list__not__last,axiom,
! [Y: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ Xs ) )
=> ( ( Y
!= ( last_set_a @ Xs ) )
=> ? [As2: list_set_a,Bs2: list_set_a] :
( ( ( append_set_a @ As2 @ ( cons_set_a @ Y @ Bs2 ) )
= Xs )
& ( Bs2 != nil_set_a ) ) ) ) ).
% split_list_not_last
thf(fact_629_split__list__not__last,axiom,
! [Y: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ Xs ) )
=> ( ( Y
!= ( last_list_a @ Xs ) )
=> ? [As2: list_list_a,Bs2: list_list_a] :
( ( ( append_list_a @ As2 @ ( cons_list_a @ Y @ Bs2 ) )
= Xs )
& ( Bs2 != nil_list_a ) ) ) ) ).
% split_list_not_last
thf(fact_630_split__list__not__last,axiom,
! [Y: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ( ( Y
!= ( last_a @ Xs ) )
=> ? [As2: list_a,Bs2: list_a] :
( ( ( append_a @ As2 @ ( cons_a @ Y @ Bs2 ) )
= Xs )
& ( Bs2 != nil_a ) ) ) ) ).
% split_list_not_last
thf(fact_631_split__list__not__last,axiom,
! [Y: b,Xs: list_b] :
( ( member_b @ Y @ ( set_b2 @ Xs ) )
=> ( ( Y
!= ( last_b @ Xs ) )
=> ? [As2: list_b,Bs2: list_b] :
( ( ( append_b @ As2 @ ( cons_b @ Y @ Bs2 ) )
= Xs )
& ( Bs2 != nil_b ) ) ) ) ).
% split_list_not_last
thf(fact_632_pre__digraph_Oawalk__verts__conv,axiom,
( arc_pr7493981781705774525ts_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,U4: a,P4: list_a] : ( if_list_a @ ( P4 = nil_a ) @ ( cons_a @ U4 @ nil_a ) @ ( append_a @ ( map_a_a @ ( pre_ta980714981981074249t_unit @ G2 ) @ P4 ) @ ( cons_a @ ( pre_he1285395828689812537t_unit @ G2 @ ( last_a @ P4 ) ) @ nil_a ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv
thf(fact_633_pre__digraph_Oawalk__verts__conv,axiom,
( arc_pr4706526199733098492ts_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,U4: b,P4: list_a] : ( if_list_b @ ( P4 = nil_a ) @ ( cons_b @ U4 @ nil_b ) @ ( append_b @ ( map_a_b @ ( pre_ta6449336744848955850t_unit @ G2 ) @ P4 ) @ ( cons_b @ ( pre_he6754017591557694138t_unit @ G2 @ ( last_a @ P4 ) ) @ nil_b ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv
thf(fact_634_pre__digraph_Oawalk__verts__conv,axiom,
( arc_pr4706526199733098493ts_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,U4: b,P4: list_b] : ( if_list_b @ ( P4 = nil_b ) @ ( cons_b @ U4 @ nil_b ) @ ( append_b @ ( map_b_b @ ( pre_ta1176856343612768521t_unit @ G2 ) @ P4 ) @ ( cons_b @ ( pre_he1481537190321506809t_unit @ G2 @ ( last_b @ P4 ) ) @ nil_b ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv
thf(fact_635_pre__digraph_Oawalk__verts__conv,axiom,
( arc_pr7493981781705774526ts_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P4: list_b] : ( if_list_a @ ( P4 = nil_b ) @ ( cons_a @ U4 @ nil_a ) @ ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ G2 ) @ P4 ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ G2 @ ( last_b @ P4 ) ) @ nil_a ) ) ) ) ) ).
% pre_digraph.awalk_verts_conv
thf(fact_636_reachable__vpath__conv,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
= ( ? [P4: list_a] :
( ( vertex_vpath_a_b @ P4 @ t )
& ( ( hd_a @ P4 )
= U2 )
& ( ( last_a @ P4 )
= V2 ) ) ) ) ).
% reachable_vpath_conv
thf(fact_637_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,U4: a,P4: list_a,V5: a] :
( ( arc_pre_trail_a_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_a2 @ P4 )
= ( pre_ar6668445444069714712t_unit @ G2 ) )
& ( ( set_a2 @ ( arc_pr7493981781705774525ts_a_a @ G2 @ U4 @ P4 ) )
= ( pre_ve5914862431884959581t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_638_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,U4: b,P4: list_a,V5: b] :
( ( arc_pre_trail_b_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_a2 @ P4 )
= ( pre_ar2913695170082820505t_unit @ G2 ) )
& ( ( set_b2 @ ( arc_pr4706526199733098492ts_b_a @ G2 @ U4 @ P4 ) )
= ( pre_ve2160112157898065374t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_639_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu4033079881512885386st_a_a
= ( ^ [G2: pre_pr8155351583225888586t_unit,U4: list_a,P4: list_a,V5: list_a] :
( ( arc_pr7309874995902050715st_a_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_a2 @ P4 )
= ( pre_ar8733286783787486494t_unit @ G2 ) )
& ( ( set_list_a2 @ ( arc_pr6350002437206376375st_a_a @ G2 @ U4 @ P4 ) )
= ( pre_ve7102540449451629283t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_640_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,U4: b,P4: list_b,V5: b] :
( ( arc_pre_trail_b_b @ G2 @ U4 @ P4 @ V5 )
& ( ( set_b2 @ P4 )
= ( pre_ar6864586805701408984t_unit @ G2 ) )
& ( ( set_b2 @ ( arc_pr4706526199733098493ts_b_b @ G2 @ U4 @ P4 ) )
= ( pre_ve6111003793516653853t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_641_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu5922315637342151574list_a
= ( ^ [G2: pre_pr3711252390037155390t_unit,U4: a,P4: list_list_a,V5: a] :
( ( arc_pr9199110751731316903list_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_list_a2 @ P4 )
= ( pre_ar4649202011430411538t_unit @ G2 ) )
& ( ( set_a2 @ ( arc_pr8239238193035642563list_a @ G2 @ U4 @ P4 ) )
= ( pre_ve3018455677094554327t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_642_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu7410277674636950101list_a
= ( ^ [G2: pre_pr7651200976661991615t_unit,U4: b,P4: list_list_a,V5: b] :
( ( arc_pr1463700752171339622list_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_list_a2 @ P4 )
= ( pre_ar405773122724138771t_unit @ G2 ) )
& ( ( set_b2 @ ( arc_pr503828193475665282list_a @ G2 @ U4 @ P4 ) )
= ( pre_ve7998398825243057368t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_643_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu6772804086596076304list_a
= ( ^ [G2: pre_pr1927278062479000516t_unit,U4: list_a,P4: list_list_a,V5: list_a] :
( ( arc_pr8660351327756884897list_a @ G2 @ U4 @ P4 @ V5 )
& ( ( set_list_a2 @ P4 )
= ( pre_ar3225631637072446488t_unit @ G2 ) )
& ( ( set_list_a2 @ ( arc_pr5757367447804505021list_a @ G2 @ U4 @ P4 ) )
= ( pre_ve2103353244832951133t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_644_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu4033079881512885387st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P4: list_b,V5: list_a] :
( ( arc_pr7309874995902050716st_a_b @ G2 @ U4 @ P4 @ V5 )
& ( ( set_b2 @ P4 )
= ( pre_ar3460806382551299165t_unit @ G2 ) )
& ( ( set_list_a2 @ ( arc_pr6350002437206376376st_a_b @ G2 @ U4 @ P4 ) )
= ( pre_ve1830060048215441954t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_645_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P4: list_b,V5: a] :
( ( arc_pre_trail_a_b @ G2 @ U4 @ P4 @ V5 )
& ( ( set_b2 @ P4 )
= ( pre_ar1395965042833527383t_unit @ G2 ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U4 @ P4 ) )
= ( pre_ve642382030648772252t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_646_awalk__verts__append2,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) )
= ( append_a @ ( butlast_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q ) ) ) ) ).
% awalk_verts_append2
thf(fact_647_awalk__verts__conv,axiom,
! [P2: list_b,U2: a] :
( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( cons_a @ U2 @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 )
= ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ ( last_b @ P2 ) ) @ nil_a ) ) ) ) ) ).
% awalk_verts_conv
thf(fact_648_awlast__of__awalk,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( nOMATCH_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ V2 )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= V2 ) ) ) ).
% awlast_of_awalk
thf(fact_649_connected__minimal,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ~ ( reachable_a_b @ ( pre_del_arc_a_b @ t @ E ) @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% connected_minimal
thf(fact_650_del__arc__commute,axiom,
! [B2: b,A2: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ B2 ) @ A2 )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ B2 ) ) ).
% del_arc_commute
thf(fact_651_del__arc__in,axiom,
! [A2: b] :
( ~ ( member_b @ A2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ t @ A2 )
= t ) ) ).
% del_arc_in
thf(fact_652_inner__verts__conv,axiom,
! [P2: list_b,U2: a] :
( ( pre_inner_verts_a_b @ t @ P2 )
= ( butlast_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% inner_verts_conv
thf(fact_653_del__del__arc__collapse,axiom,
! [A2: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ A2 )
= ( pre_del_arc_a_b @ t @ A2 ) ) ).
% del_del_arc_collapse
thf(fact_654_List_Obutlast__rev,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( rev_a @ Xs ) )
= ( rev_a @ ( tl_a @ Xs ) ) ) ).
% List.butlast_rev
thf(fact_655_List_Obutlast__rev,axiom,
! [Xs: list_b] :
( ( butlast_b @ ( rev_b @ Xs ) )
= ( rev_b @ ( tl_b @ Xs ) ) ) ).
% List.butlast_rev
thf(fact_656_verts__del__arc,axiom,
! [A2: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ t @ A2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% verts_del_arc
thf(fact_657_head__del__arc,axiom,
! [A2: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ t @ A2 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_arc
thf(fact_658_tail__del__arc,axiom,
! [A2: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ t @ A2 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_arc
thf(fact_659_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_660_butlast__snoc,axiom,
! [Xs: list_b,X: b] :
( ( butlast_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_661_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_662_append__butlast__last__id,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( ( append_b @ ( butlast_b @ Xs ) @ ( cons_b @ ( last_b @ Xs ) @ nil_b ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_663_in__set__butlastD,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( butlas8142365730073264249od_a_a @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_664_in__set__butlastD,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ ( butlast_set_a @ Xs ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_665_in__set__butlastD,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_666_in__set__butlastD,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ ( butlast_b @ Xs ) ) )
=> ( member_b @ X @ ( set_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_667_in__set__butlastD,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_668_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_669_butlast_Osimps_I1_J,axiom,
( ( butlast_b @ nil_b )
= nil_b ) ).
% butlast.simps(1)
thf(fact_670_distinct__butlast,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( butlast_a @ Xs ) ) ) ).
% distinct_butlast
thf(fact_671_distinct__butlast,axiom,
! [Xs: list_b] :
( ( distinct_b @ Xs )
=> ( distinct_b @ ( butlast_b @ Xs ) ) ) ).
% distinct_butlast
thf(fact_672_map__butlast,axiom,
! [F: a > a,Xs: list_a] :
( ( map_a_a @ F @ ( butlast_a @ Xs ) )
= ( butlast_a @ ( map_a_a @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_673_map__butlast,axiom,
! [F: b > a,Xs: list_b] :
( ( map_b_a @ F @ ( butlast_b @ Xs ) )
= ( butlast_a @ ( map_b_a @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_674_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_675_butlast__tl,axiom,
! [Xs: list_b] :
( ( butlast_b @ ( tl_b @ Xs ) )
= ( tl_b @ ( butlast_b @ Xs ) ) ) ).
% butlast_tl
thf(fact_676_in__set__butlast__appendI,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( butlas8142365730073264249od_a_a @ Xs ) ) )
| ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( butlas8142365730073264249od_a_a @ Ys ) ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( butlas8142365730073264249od_a_a @ ( append5335208819046833346od_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_677_in__set__butlast__appendI,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( ( member_set_a @ X @ ( set_set_a2 @ ( butlast_set_a @ Xs ) ) )
| ( member_set_a @ X @ ( set_set_a2 @ ( butlast_set_a @ Ys ) ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ ( butlast_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_678_in__set__butlast__appendI,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_679_in__set__butlast__appendI,axiom,
! [X: b,Xs: list_b,Ys: list_b] :
( ( ( member_b @ X @ ( set_b2 @ ( butlast_b @ Xs ) ) )
| ( member_b @ X @ ( set_b2 @ ( butlast_b @ Ys ) ) ) )
=> ( member_b @ X @ ( set_b2 @ ( butlast_b @ ( append_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_680_in__set__butlast__appendI,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
| ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Ys ) ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ ( append_list_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_681_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_682_butlast_Osimps_I2_J,axiom,
! [Xs: list_b,X: b] :
( ( ( Xs = nil_b )
=> ( ( butlast_b @ ( cons_b @ X @ Xs ) )
= nil_b ) )
& ( ( Xs != nil_b )
=> ( ( butlast_b @ ( cons_b @ X @ Xs ) )
= ( cons_b @ X @ ( butlast_b @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_683_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_684_butlast__append,axiom,
! [Ys: list_b,Xs: list_b] :
( ( ( Ys = nil_b )
=> ( ( butlast_b @ ( append_b @ Xs @ Ys ) )
= ( butlast_b @ Xs ) ) )
& ( ( Ys != nil_b )
=> ( ( butlast_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ Xs @ ( butlast_b @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_685_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_686_snoc__eq__iff__butlast,axiom,
! [Xs: list_b,X: b,Ys: list_b] :
( ( ( append_b @ Xs @ ( cons_b @ X @ nil_b ) )
= Ys )
= ( ( Ys != nil_b )
& ( ( butlast_b @ Ys )
= Xs )
& ( ( last_b @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_687_pre__digraph_Oarc__set__balanced_Ocong,axiom,
pre_ar5931435604406180204ed_a_b = pre_ar5931435604406180204ed_a_b ).
% pre_digraph.arc_set_balanced.cong
thf(fact_688_Stuff_Obutlast__rev,axiom,
! [P2: list_a] :
( ( butlast_a @ ( rev_a @ P2 ) )
= ( rev_a @ ( tl_a @ P2 ) ) ) ).
% Stuff.butlast_rev
thf(fact_689_Stuff_Obutlast__rev,axiom,
! [P2: list_b] :
( ( butlast_b @ ( rev_b @ P2 ) )
= ( rev_b @ ( tl_b @ P2 ) ) ) ).
% Stuff.butlast_rev
thf(fact_690_euler__trail__conv__connected,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( pre_euler_trail_a_b @ t @ U2 @ P2 @ V2 )
= ( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ V2 )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% euler_trail_conv_connected
thf(fact_691_Un__subset__iff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
= ( ( ord_less_eq_set_a @ A3 @ C2 )
& ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_692_Un__subset__iff,axiom,
! [A3: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C2 )
= ( ( ord_less_eq_set_b @ A3 @ C2 )
& ( ord_less_eq_set_b @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_693_bidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A: b] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A )
= A ) )
=> ( ! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A )
!= A ) )
=> ( ! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ ( Arev @ A ) )
= A ) )
=> ( ! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ ( Arev @ A ) )
= ( pre_he5236287464308401016t_unit @ t @ A ) ) )
=> ( bidire6463457107099887885ph_a_b @ t @ Arev ) ) ) ) ) ).
% bidirected_digraphI
thf(fact_694_apath__Cons__iff,axiom,
! [U2: a,E: b,Es: list_b,W: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U2 )
& ( arc_pre_apath_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es @ W )
& ~ ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es ) ) ) ) ) ).
% apath_Cons_iff
thf(fact_695_connected,axiom,
digrap8783888973171253482ed_a_b @ t ).
% connected
thf(fact_696_apath__if__awalk,axiom,
! [R2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ R2 @ P2 @ V2 )
=> ( arc_pre_apath_a_b @ t @ R2 @ P2 @ V2 ) ) ).
% apath_if_awalk
thf(fact_697_awalkI__apath,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 ) ) ).
% awalkI_apath
thf(fact_698_reachable__apath,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
= ( ? [P4: list_b] : ( arc_pre_apath_a_b @ t @ U2 @ P4 @ V2 ) ) ) ).
% reachable_apath
thf(fact_699_apath__ends,axiom,
! [U2: a,P2: list_b,V2: a,U3: a,V4: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_apath_a_b @ t @ U3 @ P2 @ V4 )
=> ( ( ( P2 != nil_b )
& ( U2 != V2 )
& ( U2 = U3 )
& ( V2 = V4 ) )
| ( ( P2 = nil_b )
& ( U2 = V2 )
& ( U3 = V4 ) ) ) ) ) ).
% apath_ends
thf(fact_700_apath__nonempty__ends,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( P2 != nil_b )
=> ( U2 != V2 ) ) ) ).
% apath_nonempty_ends
thf(fact_701_euler__imp__connected,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( pre_euler_trail_a_b @ t @ U2 @ P2 @ V2 )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% euler_imp_connected
thf(fact_702_subset__antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_703_subset__antisym,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_704_subsetI,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( member1426531477525435216od_a_a @ X2 @ B3 ) )
=> ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_705_subsetI,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A3 )
=> ( member_list_a @ X2 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_706_subsetI,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_707_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_708_subsetI,axiom,
! [A3: set_b,B3: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( member_b @ X2 @ B3 ) )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% subsetI
thf(fact_709_hd__in__awalk__verts_I2_J,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( member_a @ U2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% hd_in_awalk_verts(2)
thf(fact_710_apath__Nil__iff,axiom,
! [U2: a,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ nil_b @ V2 )
= ( ( U2 = V2 )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% apath_Nil_iff
thf(fact_711_apath__awalk__to__apath,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( arc_pre_apath_a_b @ t @ U2 @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) @ V2 ) ) ).
% apath_awalk_to_apath
thf(fact_712_unique__apath__verts__in__awalk,axiom,
! [X: a,U2: a,P1: list_b,V2: a,P22: list_b] :
( ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P1 ) ) )
=> ( ( arc_pre_apath_a_b @ t @ U2 @ P1 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P22 @ V2 )
=> ( ? [X5: list_b] :
( ( arc_pre_apath_a_b @ t @ U2 @ X5 @ V2 )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ t @ U2 @ Y2 @ V2 )
=> ( Y2 = X5 ) ) )
=> ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P22 ) ) ) ) ) ) ) ).
% unique_apath_verts_in_awalk
thf(fact_713_apath__def,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
= ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ).
% apath_def
thf(fact_714_no__loops__in__apath,axiom,
! [U2: a,P2: list_b,V2: a,A2: b] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( member_b @ A2 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ A2 )
!= ( pre_he5236287464308401016t_unit @ t @ A2 ) ) ) ) ).
% no_loops_in_apath
thf(fact_715_unique__apath__verts__sub__awalk,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ Q @ V2 )
=> ( ? [X5: list_b] :
( ( arc_pre_apath_a_b @ t @ U2 @ X5 @ V2 )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ t @ U2 @ Y2 @ V2 )
=> ( Y2 = X5 ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q ) ) ) ) ) ) ).
% unique_apath_verts_sub_awalk
thf(fact_716_apath__decomp__disjoint,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b,R2: list_b,X: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( P2
= ( append_b @ Q @ R2 ) )
=> ( ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q ) ) )
=> ~ ( member_a @ X @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q ) ) @ R2 ) ) ) ) ) ) ) ).
% apath_decomp_disjoint
thf(fact_717_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ t )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% spanning_tree_imp_connected
thf(fact_718_connected__spanning__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ t )
=> ( ( digrap8783888973171253482ed_a_b @ H )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% connected_spanning_imp_connected
thf(fact_719_pre__digraph_Oapath_Ocong,axiom,
arc_pre_apath_a_b = arc_pre_apath_a_b ).
% pre_digraph.apath.cong
thf(fact_720_pre__digraph_OawalkI__apath,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ G @ U2 @ P2 @ V2 )
=> ( arc_pre_awalk_a_b @ G @ U2 @ P2 @ V2 ) ) ).
% pre_digraph.awalkI_apath
thf(fact_721_pre__digraph_OawalkI__apath,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a,P2: list_b,V2: list_a] :
( ( arc_pr85741862633711036st_a_b @ G @ U2 @ P2 @ V2 )
=> ( arc_pr6214585750886380800st_a_b @ G @ U2 @ P2 @ V2 ) ) ).
% pre_digraph.awalkI_apath
thf(fact_722_pre__digraph_Oapath__def,axiom,
( arc_pr85741862633711036st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P4: list_b,V5: list_a] :
( ( arc_pr6214585750886380800st_a_b @ G2 @ U4 @ P4 @ V5 )
& ( distinct_list_a @ ( arc_pr6350002437206376376st_a_b @ G2 @ U4 @ P4 ) ) ) ) ) ).
% pre_digraph.apath_def
thf(fact_723_pre__digraph_Oapath__def,axiom,
( arc_pre_apath_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P4: list_b,V5: a] :
( ( arc_pre_awalk_a_b @ G2 @ U4 @ P4 @ V5 )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U4 @ P4 ) ) ) ) ) ).
% pre_digraph.apath_def
thf(fact_724_Collect__mono__iff,axiom,
! [P: a > $o,Q3: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q3 ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q3 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_725_Collect__mono__iff,axiom,
! [P: b > $o,Q3: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q3 ) )
= ( ! [X4: b] :
( ( P @ X4 )
=> ( Q3 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_726_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z6: set_a] : ( Y5 = Z6 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_727_set__eq__subset,axiom,
( ( ^ [Y5: set_b,Z6: set_b] : ( Y5 = Z6 ) )
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A4 @ B4 )
& ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_728_subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_729_subset__trans,axiom,
! [A3: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C2 )
=> ( ord_less_eq_set_b @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_730_Collect__mono,axiom,
! [P: a > $o,Q3: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q3 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q3 ) ) ) ).
% Collect_mono
thf(fact_731_Collect__mono,axiom,
! [P: b > $o,Q3: b > $o] :
( ! [X2: b] :
( ( P @ X2 )
=> ( Q3 @ X2 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q3 ) ) ) ).
% Collect_mono
thf(fact_732_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_733_subset__refl,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ A3 @ A3 ) ).
% subset_refl
thf(fact_734_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
! [T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ A4 )
=> ( member1426531477525435216od_a_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_735_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [T3: list_a] :
( ( member_list_a @ T3 @ A4 )
=> ( member_list_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_736_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A4 )
=> ( member_set_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_737_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_738_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A4 )
=> ( member_b @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_739_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_740_equalityD2,axiom,
! [A3: set_b,B3: set_b] :
( ( A3 = B3 )
=> ( ord_less_eq_set_b @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_741_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_742_equalityD1,axiom,
! [A3: set_b,B3: set_b] :
( ( A3 = B3 )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_743_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A4 )
=> ( member1426531477525435216od_a_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_744_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [X4: list_a] :
( ( member_list_a @ X4 @ A4 )
=> ( member_list_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_745_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( member_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_746_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_747_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [X4: b] :
( ( member_b @ X4 @ A4 )
=> ( member_b @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_748_equalityE,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_749_equalityE,axiom,
! [A3: set_b,B3: set_b] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_b @ A3 @ B3 )
=> ~ ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_750_subsetD,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
=> ( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_751_subsetD,axiom,
! [A3: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_list_a @ C @ A3 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_752_subsetD,axiom,
! [A3: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_753_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_754_subsetD,axiom,
! [A3: set_b,B3: set_b,C: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ C @ A3 )
=> ( member_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_755_in__mono,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
=> ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_756_in__mono,axiom,
! [A3: set_list_a,B3: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_list_a @ X @ A3 )
=> ( member_list_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_757_in__mono,axiom,
! [A3: set_set_a,B3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_758_in__mono,axiom,
! [A3: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_759_in__mono,axiom,
! [A3: set_b,B3: set_b,X: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ X @ A3 )
=> ( member_b @ X @ B3 ) ) ) ).
% in_mono
thf(fact_760_list__exhaust2,axiom,
! [Y: list_a,Ya: list_a] :
( ( ( Y = nil_a )
=> ( Ya != nil_a ) )
=> ( ( ( Y = nil_a )
=> ! [X212: a,X222: list_a] :
( Ya
!= ( cons_a @ X212 @ X222 ) ) )
=> ( ( ? [X212: a,X222: list_a] :
( Y
= ( cons_a @ X212 @ X222 ) )
=> ( Ya != nil_a ) )
=> ~ ( ? [X212: a,X222: list_a] :
( Y
= ( cons_a @ X212 @ X222 ) )
=> ! [X21a: a,X22a: list_a] :
( Ya
!= ( cons_a @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_761_list__exhaust2,axiom,
! [Y: list_b,Ya: list_a] :
( ( ( Y = nil_b )
=> ( Ya != nil_a ) )
=> ( ( ( Y = nil_b )
=> ! [X212: a,X222: list_a] :
( Ya
!= ( cons_a @ X212 @ X222 ) ) )
=> ( ( ? [X212: b,X222: list_b] :
( Y
= ( cons_b @ X212 @ X222 ) )
=> ( Ya != nil_a ) )
=> ~ ( ? [X212: b,X222: list_b] :
( Y
= ( cons_b @ X212 @ X222 ) )
=> ! [X21a: a,X22a: list_a] :
( Ya
!= ( cons_a @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_762_list__exhaust2,axiom,
! [Y: list_a,Ya: list_b] :
( ( ( Y = nil_a )
=> ( Ya != nil_b ) )
=> ( ( ( Y = nil_a )
=> ! [X212: b,X222: list_b] :
( Ya
!= ( cons_b @ X212 @ X222 ) ) )
=> ( ( ? [X212: a,X222: list_a] :
( Y
= ( cons_a @ X212 @ X222 ) )
=> ( Ya != nil_b ) )
=> ~ ( ? [X212: a,X222: list_a] :
( Y
= ( cons_a @ X212 @ X222 ) )
=> ! [X21a: b,X22a: list_b] :
( Ya
!= ( cons_b @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_763_list__exhaust2,axiom,
! [Y: list_b,Ya: list_b] :
( ( ( Y = nil_b )
=> ( Ya != nil_b ) )
=> ( ( ( Y = nil_b )
=> ! [X212: b,X222: list_b] :
( Ya
!= ( cons_b @ X212 @ X222 ) ) )
=> ( ( ? [X212: b,X222: list_b] :
( Y
= ( cons_b @ X212 @ X222 ) )
=> ( Ya != nil_b ) )
=> ~ ( ? [X212: b,X222: list_b] :
( Y
= ( cons_b @ X212 @ X222 ) )
=> ! [X21a: b,X22a: list_b] :
( Ya
!= ( cons_b @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_764_list__exhaust__NSC,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X2: a] :
( Xs
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y2: a,Ys2: list_a] :
( Xs
!= ( cons_a @ X2 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust_NSC
thf(fact_765_list__exhaust__NSC,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( ! [X2: b] :
( Xs
!= ( cons_b @ X2 @ nil_b ) )
=> ~ ! [X2: b,Y2: b,Ys2: list_b] :
( Xs
!= ( cons_b @ X2 @ ( cons_b @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust_NSC
thf(fact_766_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_767_subset__Un__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( sup_sup_set_b @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_768_subset__UnE,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A3 )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B3 )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_769_subset__UnE,axiom,
! [C2: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) )
=> ~ ! [A5: set_b] :
( ( ord_less_eq_set_b @ A5 @ A3 )
=> ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ B3 )
=> ( C2
!= ( sup_sup_set_b @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_770_Un__absorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_771_Un__absorb2,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( sup_sup_set_b @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_772_Un__absorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_773_Un__absorb1,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( sup_sup_set_b @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_774_Un__upper2,axiom,
! [B3: set_a,A3: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_775_Un__upper2,axiom,
! [B3: set_b,A3: set_b] : ( ord_less_eq_set_b @ B3 @ ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_776_Un__upper1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_777_Un__upper1,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ A3 @ ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_778_Un__least,axiom,
! [A3: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_779_Un__least,axiom,
! [A3: set_b,C2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ C2 )
=> ( ( ord_less_eq_set_b @ B3 @ C2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_780_Un__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_781_Un__mono,axiom,
! [A3: set_b,C2: set_b,B3: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A3 @ C2 )
=> ( ( ord_less_eq_set_b @ B3 @ D )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ ( sup_sup_set_b @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_782_list__set__tl,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_783_list__set__tl,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_784_list__set__tl,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_785_list__set__tl,axiom,
! [X: b,Xs: list_b] :
( ( member_b @ X @ ( set_b2 @ ( tl_b @ Xs ) ) )
=> ( member_b @ X @ ( set_b2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_786_list__set__tl,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( tl_list_a @ Xs ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_787_tl__rev,axiom,
! [P2: list_a] :
( ( tl_a @ ( rev_a @ P2 ) )
= ( rev_a @ ( butlast_a @ P2 ) ) ) ).
% tl_rev
thf(fact_788_tl__rev,axiom,
! [P2: list_b] :
( ( tl_b @ ( rev_b @ P2 ) )
= ( rev_b @ ( butlast_b @ P2 ) ) ) ).
% tl_rev
thf(fact_789_not__distinct__decomp__min__prefix,axiom,
! [Ws: list_P1396940483166286381od_a_a] :
( ~ ( distin132333870042060960od_a_a @ Ws )
=> ? [Xs2: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a,Y2: product_prod_a_a] :
( ( Ws
= ( append5335208819046833346od_a_a @ Xs2 @ ( cons_P7316939126706565853od_a_a @ Y2 @ ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ Y2 @ Zs3 ) ) ) ) )
& ( distin132333870042060960od_a_a @ Xs2 )
& ~ ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ~ ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Ys2 ) ) ) ) ).
% not_distinct_decomp_min_prefix
thf(fact_790_not__distinct__decomp__min__prefix,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs2: list_set_a,Ys2: list_set_a,Zs3: list_set_a,Y2: set_a] :
( ( Ws
= ( append_set_a @ Xs2 @ ( cons_set_a @ Y2 @ ( append_set_a @ Ys2 @ ( cons_set_a @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_set_a @ Xs2 )
& ~ ( member_set_a @ Y2 @ ( set_set_a2 @ Xs2 ) )
& ~ ( member_set_a @ Y2 @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% not_distinct_decomp_min_prefix
thf(fact_791_not__distinct__decomp__min__prefix,axiom,
! [Ws: list_list_a] :
( ~ ( distinct_list_a @ Ws )
=> ? [Xs2: list_list_a,Ys2: list_list_a,Zs3: list_list_a,Y2: list_a] :
( ( Ws
= ( append_list_a @ Xs2 @ ( cons_list_a @ Y2 @ ( append_list_a @ Ys2 @ ( cons_list_a @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_list_a @ Xs2 )
& ~ ( member_list_a @ Y2 @ ( set_list_a2 @ Xs2 ) )
& ~ ( member_list_a @ Y2 @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% not_distinct_decomp_min_prefix
thf(fact_792_not__distinct__decomp__min__prefix,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs3: list_a,Y2: a] :
( ( Ws
= ( append_a @ Xs2 @ ( cons_a @ Y2 @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_a @ Xs2 )
& ~ ( member_a @ Y2 @ ( set_a2 @ Xs2 ) )
& ~ ( member_a @ Y2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% not_distinct_decomp_min_prefix
thf(fact_793_not__distinct__decomp__min__prefix,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs2: list_b,Ys2: list_b,Zs3: list_b,Y2: b] :
( ( Ws
= ( append_b @ Xs2 @ ( cons_b @ Y2 @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_b @ Xs2 )
& ~ ( member_b @ Y2 @ ( set_b2 @ Xs2 ) )
& ~ ( member_b @ Y2 @ ( set_b2 @ Ys2 ) ) ) ) ).
% not_distinct_decomp_min_prefix
thf(fact_794_not__distinct__decomp__min__not__distinct,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Y2: a,Ys2: list_a,Zs3: list_a] :
( ( Ws
= ( append_a @ Xs2 @ ( cons_a @ Y2 @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_a @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ) ).
% not_distinct_decomp_min_not_distinct
thf(fact_795_not__distinct__decomp__min__not__distinct,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs2: list_b,Y2: b,Ys2: list_b,Zs3: list_b] :
( ( Ws
= ( append_b @ Xs2 @ ( cons_b @ Y2 @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ Zs3 ) ) ) ) )
& ( distinct_b @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ) ).
% not_distinct_decomp_min_not_distinct
thf(fact_796_trail__connected,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_trail_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ) ) ).
% trail_connected
thf(fact_797_awalk__connected,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ) ) ).
% awalk_connected
thf(fact_798_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_799_sup_Obounded__iff,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_b @ B2 @ A2 )
& ( ord_less_eq_set_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_800_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_801_le__sup__iff,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_b @ X @ Z )
& ( ord_less_eq_set_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_802_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ! [Xa3: a] :
( ( member_a @ Xa3 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ X2 @ Xa3 ) ) )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% verts_reachable_connected
thf(fact_803_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_804_merging__empty,axiom,
( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ).
% merging_empty
thf(fact_805_in__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( ( S != bot_bot_set_a )
& ! [X4: a] :
( ( member_a @ X4 @ S )
=> ! [Y3: a] :
( ( member_a @ Y3 @ S )
=> ( reachable_a_b @ t @ X4 @ Y3 ) ) )
& ! [X4: a] :
( ( member_a @ X4 @ S )
=> ! [V5: a] :
( ~ ( member_a @ V5 @ S )
=> ( ~ ( reachable_a_b @ t @ X4 @ V5 )
| ~ ( reachable_a_b @ t @ V5 @ X4 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_806_scc__of__empty__conv,axiom,
! [U2: a] :
( ( ( digrap2937667069914300949of_a_b @ t @ U2 )
= bot_bot_set_a )
= ( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% scc_of_empty_conv
thf(fact_807_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_808_subset__empty,axiom,
! [A3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ bot_bo1839476491465656141t_unit )
= ( A3 = bot_bo1839476491465656141t_unit ) ) ).
% subset_empty
thf(fact_809_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_810_subset__empty,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_811_empty__subsetI,axiom,
! [A3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A3 ) ).
% empty_subsetI
thf(fact_812_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_813_empty__subsetI,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A3 ) ).
% empty_subsetI
thf(fact_814_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_815_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_816_set__empty,axiom,
! [Xs: list_b] :
( ( ( set_b2 @ Xs )
= bot_bot_set_b )
= ( Xs = nil_b ) ) ).
% set_empty
thf(fact_817_set__empty,axiom,
! [Xs: list_p1584440430088372499t_unit] :
( ( ( set_pr4006367424059803566t_unit @ Xs )
= bot_bo1839476491465656141t_unit )
= ( Xs = nil_pr1362588839662627709t_unit ) ) ).
% set_empty
thf(fact_818_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_819_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_820_set__empty2,axiom,
! [Xs: list_b] :
( ( bot_bot_set_b
= ( set_b2 @ Xs ) )
= ( Xs = nil_b ) ) ).
% set_empty2
thf(fact_821_set__empty2,axiom,
! [Xs: list_p1584440430088372499t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( set_pr4006367424059803566t_unit @ Xs ) )
= ( Xs = nil_pr1362588839662627709t_unit ) ) ).
% set_empty2
thf(fact_822_Diff__eq__empty__iff,axiom,
! [A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( minus_3777555517894451474t_unit @ A3 @ B3 )
= bot_bo1839476491465656141t_unit )
= ( ord_le8200006823705900825t_unit @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_823_Diff__eq__empty__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_824_Diff__eq__empty__iff,axiom,
! [A3: set_b,B3: set_b] :
( ( ( minus_minus_set_b @ A3 @ B3 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_825_Diff__mono,axiom,
! [A3: set_a,C2: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_826_Diff__mono,axiom,
! [A3: set_b,C2: set_b,D: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ C2 )
=> ( ( ord_less_eq_set_b @ D @ B3 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( minus_minus_set_b @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_827_Diff__subset,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_828_Diff__subset,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_829_double__diff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_830_double__diff,axiom,
! [A3: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C2 )
=> ( ( minus_minus_set_b @ B3 @ ( minus_minus_set_b @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_831_Diff__partition,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_832_Diff__partition,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( sup_sup_set_b @ A3 @ ( minus_minus_set_b @ B3 @ A3 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_833_Diff__subset__conv,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ C2 )
= ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_834_Diff__subset__conv,axiom,
! [A3: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ C2 )
= ( ord_less_eq_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_835_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_836_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_837_empty__set,axiom,
( bot_bot_set_b
= ( set_b2 @ nil_b ) ) ).
% empty_set
thf(fact_838_empty__set,axiom,
( bot_bo1839476491465656141t_unit
= ( set_pr4006367424059803566t_unit @ nil_pr1362588839662627709t_unit ) ) ).
% empty_set
thf(fact_839_distinct__set__diff,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( append_list_a @ Xs @ Ys ) )
=> ( ( set_list_a2 @ Ys )
= ( minus_646659088055828811list_a @ ( set_list_a2 @ ( append_list_a @ Xs @ Ys ) ) @ ( set_list_a2 @ Xs ) ) ) ) ).
% distinct_set_diff
thf(fact_840_distinct__set__diff,axiom,
! [Xs: list_b,Ys: list_b] :
( ( distinct_b @ ( append_b @ Xs @ Ys ) )
=> ( ( set_b2 @ Ys )
= ( minus_minus_set_b @ ( set_b2 @ ( append_b @ Xs @ Ys ) ) @ ( set_b2 @ Xs ) ) ) ) ).
% distinct_set_diff
thf(fact_841_distinct__set__diff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs @ Ys ) )
=> ( ( set_a2 @ Ys )
= ( minus_minus_set_a @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) @ ( set_a2 @ Xs ) ) ) ) ).
% distinct_set_diff
thf(fact_842_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_843_inf__sup__ord_I4_J,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_844_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_845_inf__sup__ord_I3_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_846_le__supE,axiom,
! [A2: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_847_le__supE,axiom,
! [A2: set_b,B2: set_b,X: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_b @ A2 @ X )
=> ~ ( ord_less_eq_set_b @ B2 @ X ) ) ) ).
% le_supE
thf(fact_848_le__supI,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_849_le__supI,axiom,
! [A2: set_b,X: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ X )
=> ( ( ord_less_eq_set_b @ B2 @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_850_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_851_sup__ge1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_852_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_853_sup__ge2,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_854_le__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_855_le__supI1,axiom,
! [X: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X @ A2 )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_856_le__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_857_le__supI2,axiom,
! [X: set_b,B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ X @ B2 )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_858_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_859_sup_Omono,axiom,
! [C: set_b,A2: set_b,D2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ( ord_less_eq_set_b @ D2 @ B2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ C @ D2 ) @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_860_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_861_sup__mono,axiom,
! [A2: set_b,C: set_b,B2: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B2 @ D2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ ( sup_sup_set_b @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_862_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_863_sup__least,axiom,
! [Y: set_b,X: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ Z @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_864_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_865_le__iff__sup,axiom,
( ord_less_eq_set_b
= ( ^ [X4: set_b,Y3: set_b] :
( ( sup_sup_set_b @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_866_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_867_sup_OorderE,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_868_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_869_sup_OorderI,axiom,
! [A2: set_b,B2: set_b] :
( ( A2
= ( sup_sup_set_b @ A2 @ B2 ) )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_870_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a,Z5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z5 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z5 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_871_sup__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_b,Y2: set_b,Z5: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( ( ord_less_eq_set_b @ Z5 @ X2 )
=> ( ord_less_eq_set_b @ ( F @ Y2 @ Z5 ) @ X2 ) ) )
=> ( ( sup_sup_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_872_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_873_sup_Oabsorb1,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_874_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_875_sup_Oabsorb2,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_876_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_877_sup__absorb1,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( sup_sup_set_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_878_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_879_sup__absorb2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( sup_sup_set_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_880_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_881_sup_OboundedE,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ~ ( ord_less_eq_set_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_882_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_883_sup_OboundedI,axiom,
! [B2: set_b,A2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( ord_less_eq_set_b @ C @ A2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_884_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A6: set_a] :
( A6
= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_885_sup_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [B6: set_b,A6: set_b] :
( A6
= ( sup_sup_set_b @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_886_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_887_sup_Ocobounded1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_888_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_889_sup_Ocobounded2,axiom,
! [B2: set_b,A2: set_b] : ( ord_less_eq_set_b @ B2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_890_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_891_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [B6: set_b,A6: set_b] :
( ( sup_sup_set_b @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_892_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_893_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( sup_sup_set_b @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_894_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_895_sup_OcoboundedI1,axiom,
! [C: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_896_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_897_sup_OcoboundedI2,axiom,
! [C: set_b,B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ C @ B2 )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_898_set__awalk__verts__cas,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U2 @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts_cas
thf(fact_899_set__awalk__verts,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U2 @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts
thf(fact_900_apath__append__iff,axiom,
! [U2: a,P2: list_b,Q: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ ( append_b @ P2 @ Q ) @ V2 )
= ( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) )
& ( arc_pre_apath_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q @ V2 )
& ( ( inf_inf_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ Q ) ) ) )
= bot_bot_set_a ) ) ) ).
% apath_append_iff
thf(fact_901_to__list__tree__disjoint__verts,axiom,
! [U2: list_a,V2: list_a] :
( ( member_list_a @ U2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( member_list_a @ V2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( U2 != V2 )
=> ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a ) ) ) ) ).
% to_list_tree_disjoint_verts
thf(fact_902_pre__digraph_Oinner__verts__conv,axiom,
! [U2: a] :
( pre_inner_verts_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,P4: list_b] : ( butlast_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U2 @ P4 ) ) ) ) ) ).
% pre_digraph.inner_verts_conv
thf(fact_903_verts__add__vert,axiom,
! [U2: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ t @ U2 ) )
= ( insert_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% verts_add_vert
thf(fact_904_sccs__verts__disjoint,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( member_set_a @ T @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( S != T )
=> ( ( inf_inf_set_a @ S @ T )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_905_verts__del__vert,axiom,
! [U2: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ t @ U2 ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ U2 @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_906_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ~ ! [V: a] :
( ( pre_ve642382030648772252t_unit @ t )
!= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_907_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_908_le__inf__iff,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
= ( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_909_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_910_inf_Obounded__iff,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
= ( ( ord_less_eq_set_b @ A2 @ B2 )
& ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_911_insert__subset,axiom,
! [X: pre_pr7278220950009878019t_unit,A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ X @ A3 ) @ B3 )
= ( ( member6939884229742472986t_unit @ X @ B3 )
& ( ord_le8200006823705900825t_unit @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_912_insert__subset,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A3 ) @ B3 )
= ( ( member1426531477525435216od_a_a @ X @ B3 )
& ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_913_insert__subset,axiom,
! [X: list_a,A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A3 ) @ B3 )
= ( ( member_list_a @ X @ B3 )
& ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_914_insert__subset,axiom,
! [X: set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A3 ) @ B3 )
= ( ( member_set_a @ X @ B3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_915_insert__subset,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_916_insert__subset,axiom,
! [X: b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A3 ) @ B3 )
= ( ( member_b @ X @ B3 )
& ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_917_Int__subset__iff,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_918_Int__subset__iff,axiom,
! [C2: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) )
= ( ( ord_less_eq_set_b @ C2 @ A3 )
& ( ord_less_eq_set_b @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_919_set__inner__verts,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) )
= ( minus_minus_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) @ ( insert_a @ U2 @ ( insert_a @ V2 @ bot_bot_set_a ) ) ) ) ) ).
% set_inner_verts
thf(fact_920_list_Osimps_I15_J,axiom,
! [X21: pre_pr7278220950009878019t_unit,X22: list_p1584440430088372499t_unit] :
( ( set_pr4006367424059803566t_unit @ ( cons_p8564118325392489421t_unit @ X21 @ X22 ) )
= ( insert6864688055023459379t_unit @ X21 @ ( set_pr4006367424059803566t_unit @ X22 ) ) ) ).
% list.simps(15)
thf(fact_921_list_Osimps_I15_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) )
= ( insert_list_a @ X21 @ ( set_list_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_922_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_923_list_Osimps_I15_J,axiom,
! [X21: b,X22: list_b] :
( ( set_b2 @ ( cons_b @ X21 @ X22 ) )
= ( insert_b @ X21 @ ( set_b2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_924_singleton__insert__inj__eq,axiom,
! [B2: pre_pr7278220950009878019t_unit,A2: pre_pr7278220950009878019t_unit,A3: set_pr5411798346947241657t_unit] :
( ( ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_925_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A3: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_926_singleton__insert__inj__eq,axiom,
! [B2: b,A2: b,A3: set_b] :
( ( ( insert_b @ B2 @ bot_bot_set_b )
= ( insert_b @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_927_singleton__insert__inj__eq_H,axiom,
! [A2: pre_pr7278220950009878019t_unit,A3: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A2 @ A3 )
= ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) )
= ( ( A2 = B2 )
& ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_928_singleton__insert__inj__eq_H,axiom,
! [A2: a,A3: set_a,B2: a] :
( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_929_singleton__insert__inj__eq_H,axiom,
! [A2: b,A3: set_b,B2: b] :
( ( ( insert_b @ A2 @ A3 )
= ( insert_b @ B2 @ bot_bot_set_b ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_930_distinct__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( ( distinct_list_a @ Xs )
& ( distinct_list_a @ Ys )
& ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a ) ) ) ).
% distinct_append
thf(fact_931_distinct__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_a @ Xs )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_932_distinct__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( distinct_b @ ( append_b @ Xs @ Ys ) )
= ( ( distinct_b @ Xs )
& ( distinct_b @ Ys )
& ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b ) ) ) ).
% distinct_append
thf(fact_933_distinct__append,axiom,
! [Xs: list_p1584440430088372499t_unit,Ys: list_p1584440430088372499t_unit] :
( ( distin5038060903466274250t_unit @ ( append1841976088027155496t_unit @ Xs @ Ys ) )
= ( ( distin5038060903466274250t_unit @ Xs )
& ( distin5038060903466274250t_unit @ Ys )
& ( ( inf_in1092213268631476299t_unit @ ( set_pr4006367424059803566t_unit @ Xs ) @ ( set_pr4006367424059803566t_unit @ Ys ) )
= bot_bo1839476491465656141t_unit ) ) ) ).
% distinct_append
thf(fact_934_gen__iapath__def,axiom,
! [V3: set_a,U2: a,P2: list_b,V2: a] :
( ( pre_gen_iapath_a_b @ t @ V3 @ U2 @ P2 @ V2 )
= ( ( member_a @ U2 @ V3 )
& ( member_a @ V2 @ V3 )
& ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 )
& ( ( inf_inf_set_a @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) ) @ V3 )
= bot_bot_set_a )
& ( P2 != nil_b ) ) ) ).
% gen_iapath_def
thf(fact_935_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_936_inf__sup__ord_I2_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_937_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_938_inf__sup__ord_I1_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_939_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_940_inf__le1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_941_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_942_inf__le2,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_943_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_944_le__infE,axiom,
! [X: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_b @ X @ A2 )
=> ~ ( ord_less_eq_set_b @ X @ B2 ) ) ) ).
% le_infE
thf(fact_945_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_946_le__infI,axiom,
! [X: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X @ B2 )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_947_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_948_inf__mono,axiom,
! [A2: set_b,C: set_b,B2: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B2 @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_949_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_950_le__infI1,axiom,
! [A2: set_b,X: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_951_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_952_le__infI2,axiom,
! [B2: set_b,X: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_953_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_954_inf_OorderE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_955_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_956_inf_OorderI,axiom,
! [A2: set_b,B2: set_b] :
( ( A2
= ( inf_inf_set_b @ A2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_957_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_a,Y2: set_a,Z5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z5 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y2 @ Z5 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_958_inf__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_b,Y2: set_b,Z5: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ X2 @ Z5 )
=> ( ord_less_eq_set_b @ X2 @ ( F @ Y2 @ Z5 ) ) ) )
=> ( ( inf_inf_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_959_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_960_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X4: set_b,Y3: set_b] :
( ( inf_inf_set_b @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_961_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_962_inf_Oabsorb1,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_963_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_964_inf_Oabsorb2,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_965_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_966_inf__absorb1,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( inf_inf_set_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_967_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_968_inf__absorb2,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( inf_inf_set_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_969_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_970_inf_OboundedE,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_971_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_972_inf_OboundedI,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_973_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_974_inf__greatest,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Z )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_975_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_976_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( A6
= ( inf_inf_set_b @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_977_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_978_inf_Ocobounded1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_979_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_980_inf_Ocobounded2,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_981_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( inf_inf_set_a @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_982_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( inf_inf_set_b @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_983_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_984_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B6: set_b,A6: set_b] :
( ( inf_inf_set_b @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_985_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_986_inf_OcoboundedI1,axiom,
! [A2: set_b,C: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_987_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_988_inf_OcoboundedI2,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_989_Int__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_990_Int__mono,axiom,
! [A3: set_b,C2: set_b,B3: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A3 @ C2 )
=> ( ( ord_less_eq_set_b @ B3 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_991_Int__lower1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_992_Int__lower1,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_993_Int__lower2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_994_Int__lower2,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_995_Int__absorb1,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_996_Int__absorb1,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_997_Int__absorb2,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_998_Int__absorb2,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( inf_inf_set_b @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_999_Int__greatest,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_1000_Int__greatest,axiom,
! [C2: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ A3 )
=> ( ( ord_less_eq_set_b @ C2 @ B3 )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_1001_Int__Collect__mono,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,P: product_prod_a_a > $o,Q3: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q3 @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B3 @ ( collec3336397797384452498od_a_a @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1002_Int__Collect__mono,axiom,
! [A3: set_list_a,B3: set_list_a,P: list_a > $o,Q3: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q3 @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1003_Int__Collect__mono,axiom,
! [A3: set_set_a,B3: set_set_a,P: set_a > $o,Q3: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q3 @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1004_Int__Collect__mono,axiom,
! [A3: set_a,B3: set_a,P: a > $o,Q3: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q3 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1005_Int__Collect__mono,axiom,
! [A3: set_b,B3: set_b,P: b > $o,Q3: b > $o] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q3 @ X2 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B3 @ ( collect_b @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1006_insert__mono,axiom,
! [C2: set_pr5411798346947241657t_unit,D: set_pr5411798346947241657t_unit,A2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ C2 @ D )
=> ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ A2 @ C2 ) @ ( insert6864688055023459379t_unit @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1007_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1008_insert__mono,axiom,
! [C2: set_b,D: set_b,A2: b] :
( ( ord_less_eq_set_b @ C2 @ D )
=> ( ord_less_eq_set_b @ ( insert_b @ A2 @ C2 ) @ ( insert_b @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1009_subset__insert,axiom,
! [X: pre_pr7278220950009878019t_unit,A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A3 )
=> ( ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( ord_le8200006823705900825t_unit @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1010_subset__insert,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ B3 ) )
= ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1011_subset__insert,axiom,
! [X: list_a,A3: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B3 ) )
= ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1012_subset__insert,axiom,
! [X: set_a,A3: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1013_subset__insert,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1014_subset__insert,axiom,
! [X: b,A3: set_b,B3: set_b] :
( ~ ( member_b @ X @ A3 )
=> ( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X @ B3 ) )
= ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1015_subset__insertI,axiom,
! [B3: set_pr5411798346947241657t_unit,A2: pre_pr7278220950009878019t_unit] : ( ord_le8200006823705900825t_unit @ B3 @ ( insert6864688055023459379t_unit @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1016_subset__insertI,axiom,
! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1017_subset__insertI,axiom,
! [B3: set_b,A2: b] : ( ord_less_eq_set_b @ B3 @ ( insert_b @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1018_subset__insertI2,axiom,
! [A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ B3 )
=> ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1019_subset__insertI2,axiom,
! [A3: set_a,B3: set_a,B2: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1020_subset__insertI2,axiom,
! [A3: set_b,B3: set_b,B2: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1021_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1022_distrib__sup__le,axiom,
! [X: set_b,Y: set_b,Z: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y ) @ ( sup_sup_set_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1023_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1024_distrib__inf__le,axiom,
! [X: set_b,Y: set_b,Z: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y ) @ ( inf_inf_set_b @ X @ Z ) ) @ ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1025_subset__singletonD,axiom,
! [A3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) )
=> ( ( A3 = bot_bo1839476491465656141t_unit )
| ( A3
= ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_1026_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1027_subset__singletonD,axiom,
! [A3: set_b,X: b] :
( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A3 = bot_bot_set_b )
| ( A3
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_1028_subset__singleton__iff,axiom,
! [X6: set_pr5411798346947241657t_unit,A2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ X6 @ ( insert6864688055023459379t_unit @ A2 @ bot_bo1839476491465656141t_unit ) )
= ( ( X6 = bot_bo1839476491465656141t_unit )
| ( X6
= ( insert6864688055023459379t_unit @ A2 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_1029_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1030_subset__singleton__iff,axiom,
! [X6: set_b,A2: b] :
( ( ord_less_eq_set_b @ X6 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( ( X6 = bot_bot_set_b )
| ( X6
= ( insert_b @ A2 @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1031_Un__Int__assoc__eq,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1032_Un__Int__assoc__eq,axiom,
! [A3: set_b,B3: set_b,C2: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C2 ) ) )
= ( ord_less_eq_set_b @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1033_subset__Diff__insert,axiom,
! [A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,C2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ ( minus_3777555517894451474t_unit @ B3 @ ( insert6864688055023459379t_unit @ X @ C2 ) ) )
= ( ( ord_le8200006823705900825t_unit @ A3 @ ( minus_3777555517894451474t_unit @ B3 @ C2 ) )
& ~ ( member6939884229742472986t_unit @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1034_subset__Diff__insert,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,X: product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ X @ C2 ) ) )
= ( ( ord_le746702958409616551od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B3 @ C2 ) )
& ~ ( member1426531477525435216od_a_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1035_subset__Diff__insert,axiom,
! [A3: set_list_a,B3: set_list_a,X: list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( minus_646659088055828811list_a @ B3 @ ( insert_list_a @ X @ C2 ) ) )
= ( ( ord_le8861187494160871172list_a @ A3 @ ( minus_646659088055828811list_a @ B3 @ C2 ) )
& ~ ( member_list_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1036_subset__Diff__insert,axiom,
! [A3: set_set_a,B3: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B3 @ C2 ) )
& ~ ( member_set_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1037_subset__Diff__insert,axiom,
! [A3: set_a,B3: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B3 @ C2 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1038_subset__Diff__insert,axiom,
! [A3: set_b,B3: set_b,X: b,C2: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( minus_minus_set_b @ B3 @ ( insert_b @ X @ C2 ) ) )
= ( ( ord_less_eq_set_b @ A3 @ ( minus_minus_set_b @ B3 @ C2 ) )
& ~ ( member_b @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1039_subset__insert__iff,axiom,
! [A3: set_Product_prod_a_a,X: product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ B3 ) )
= ( ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B3 ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1040_subset__insert__iff,axiom,
! [A3: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B3 ) )
= ( ( ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1041_subset__insert__iff,axiom,
! [A3: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B3 ) )
= ( ( ( member_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1042_subset__insert__iff,axiom,
! [A3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( ( ( member6939884229742472986t_unit @ X @ A3 )
=> ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A3 )
=> ( ord_le8200006823705900825t_unit @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1043_subset__insert__iff,axiom,
! [A3: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1044_subset__insert__iff,axiom,
! [A3: set_b,X: b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X @ B3 ) )
= ( ( ( member_b @ X @ A3 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 ) )
& ( ~ ( member_b @ X @ A3 )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1045_Diff__single__insert,axiom,
! [A3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 )
=> ( ord_le8200006823705900825t_unit @ A3 @ ( insert6864688055023459379t_unit @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1046_Diff__single__insert,axiom,
! [A3: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1047_Diff__single__insert,axiom,
! [A3: set_b,X: b,B3: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 )
=> ( ord_less_eq_set_b @ A3 @ ( insert_b @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1048_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( Xs = nil_list_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_1049_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_p1584440430088372499t_unit,Ys: list_p1584440430088372499t_unit] :
( ( ord_le8200006823705900825t_unit @ ( set_pr4006367424059803566t_unit @ Xs ) @ ( set_pr4006367424059803566t_unit @ Ys ) )
=> ( ( ( inf_in1092213268631476299t_unit @ ( set_pr4006367424059803566t_unit @ Xs ) @ ( set_pr4006367424059803566t_unit @ Ys ) )
= bot_bo1839476491465656141t_unit )
=> ( Xs = nil_pr1362588839662627709t_unit ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_1050_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( Xs = nil_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_1051_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b )
=> ( Xs = nil_b ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_1052_pre__digraph_Oinner__verts_Ocong,axiom,
pre_inner_verts_a_b = pre_inner_verts_a_b ).
% pre_digraph.inner_verts.cong
thf(fact_1053_pre__digraph_Oinner__verts__Nil,axiom,
! [G: pre_pr3327329314391289540t_unit] :
( ( pre_inner_verts_a_a @ G @ nil_a )
= nil_a ) ).
% pre_digraph.inner_verts_Nil
thf(fact_1054_pre__digraph_Oinner__verts__Nil,axiom,
! [G: pre_pr3994228789931197893t_unit] :
( ( pre_inner_verts_b_a @ G @ nil_a )
= nil_b ) ).
% pre_digraph.inner_verts_Nil
thf(fact_1055_pre__digraph_Oinner__verts__Nil,axiom,
! [G: pre_pr7945120425549786372t_unit] :
( ( pre_inner_verts_b_b @ G @ nil_b )
= nil_b ) ).
% pre_digraph.inner_verts_Nil
thf(fact_1056_pre__digraph_Oinner__verts__Nil,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( pre_inner_verts_a_b @ G @ nil_b )
= nil_a ) ).
% pre_digraph.inner_verts_Nil
thf(fact_1057_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: product_prod_a_a,G: pre_pr7908921069988166637t_unit,P2: list_a,Q: list_a] :
( ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487987_a_a_a @ G @ P2 ) ) )
=> ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487987_a_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1058_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: set_a,G: pre_pr3647964229410195492t_unit,P2: list_a,Q: list_a] :
( ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948156et_a_a @ G @ P2 ) ) )
=> ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948156et_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1059_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: product_prod_a_a,G: pre_pr2636440668751979308t_unit,P2: list_b,Q: list_b] :
( ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487988_a_a_b @ G @ P2 ) ) )
=> ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487988_a_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1060_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: set_a,G: pre_pr7598855865028783971t_unit,P2: list_b,Q: list_b] :
( ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948157et_a_b @ G @ P2 ) ) )
=> ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948157et_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1061_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: a,G: pre_pr3327329314391289540t_unit,P2: list_a,Q: list_a] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_a @ G @ P2 ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1062_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: b,G: pre_pr3994228789931197893t_unit,P2: list_a,Q: list_a] :
( ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_a @ G @ P2 ) ) )
=> ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1063_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: b,G: pre_pr7945120425549786372t_unit,P2: list_b,Q: list_b] :
( ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_b @ G @ P2 ) ) )
=> ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1064_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: list_a,G: pre_pr8155351583225888586t_unit,P2: list_a,Q: list_a] :
( ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915414st_a_a @ G @ P2 ) ) )
=> ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915414st_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1065_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: list_a,G: pre_pr2882871181989701257t_unit,P2: list_b,Q: list_b] :
( ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915415st_a_b @ G @ P2 ) ) )
=> ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915415st_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1066_pre__digraph_Oin__set__inner__verts__appendI__l,axiom,
! [U2: a,G: pre_pr7278220950009878019t_unit,P2: list_b,Q: list_b] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ G @ P2 ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_l
thf(fact_1067_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: product_prod_a_a,G: pre_pr7908921069988166637t_unit,Q: list_a,P2: list_a] :
( ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487987_a_a_a @ G @ Q ) ) )
=> ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487987_a_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1068_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: set_a,G: pre_pr3647964229410195492t_unit,Q: list_a,P2: list_a] :
( ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948156et_a_a @ G @ Q ) ) )
=> ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948156et_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1069_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: product_prod_a_a,G: pre_pr2636440668751979308t_unit,Q: list_b,P2: list_b] :
( ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487988_a_a_b @ G @ Q ) ) )
=> ( member1426531477525435216od_a_a @ U2 @ ( set_Product_prod_a_a2 @ ( pre_in7981346435028487988_a_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1070_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: set_a,G: pre_pr7598855865028783971t_unit,Q: list_b,P2: list_b] :
( ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948157et_a_b @ G @ Q ) ) )
=> ( member_set_a @ U2 @ ( set_set_a2 @ ( pre_in289353705076948157et_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1071_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: a,G: pre_pr3327329314391289540t_unit,Q: list_a,P2: list_a] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_a @ G @ Q ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1072_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: b,G: pre_pr3994228789931197893t_unit,Q: list_a,P2: list_a] :
( ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_a @ G @ Q ) ) )
=> ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1073_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: b,G: pre_pr7945120425549786372t_unit,Q: list_b,P2: list_b] :
( ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_b @ G @ Q ) ) )
=> ( member_b @ U2 @ ( set_b2 @ ( pre_inner_verts_b_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1074_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: list_a,G: pre_pr8155351583225888586t_unit,Q: list_a,P2: list_a] :
( ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915414st_a_a @ G @ Q ) ) )
=> ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915414st_a_a @ G @ ( append_a @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1075_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: list_a,G: pre_pr2882871181989701257t_unit,Q: list_b,P2: list_b] :
( ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915415st_a_b @ G @ Q ) ) )
=> ( member_list_a @ U2 @ ( set_list_a2 @ ( pre_in4739640915595915415st_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1076_pre__digraph_Oin__set__inner__verts__appendI__r,axiom,
! [U2: a,G: pre_pr7278220950009878019t_unit,Q: list_b,P2: list_b] :
( ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ G @ Q ) ) )
=> ( member_a @ U2 @ ( set_a2 @ ( pre_inner_verts_a_b @ G @ ( append_b @ P2 @ Q ) ) ) ) ) ).
% pre_digraph.in_set_inner_verts_appendI_r
thf(fact_1077_pre__digraph_Oinner__verts__singleton,axiom,
! [G: pre_pr3327329314391289540t_unit,X: a] :
( ( pre_inner_verts_a_a @ G @ ( cons_a @ X @ nil_a ) )
= nil_a ) ).
% pre_digraph.inner_verts_singleton
thf(fact_1078_pre__digraph_Oinner__verts__singleton,axiom,
! [G: pre_pr3994228789931197893t_unit,X: a] :
( ( pre_inner_verts_b_a @ G @ ( cons_a @ X @ nil_a ) )
= nil_b ) ).
% pre_digraph.inner_verts_singleton
thf(fact_1079_pre__digraph_Oinner__verts__singleton,axiom,
! [G: pre_pr7945120425549786372t_unit,X: b] :
( ( pre_inner_verts_b_b @ G @ ( cons_b @ X @ nil_b ) )
= nil_b ) ).
% pre_digraph.inner_verts_singleton
thf(fact_1080_pre__digraph_Oinner__verts__singleton,axiom,
! [G: pre_pr7278220950009878019t_unit,X: b] :
( ( pre_inner_verts_a_b @ G @ ( cons_b @ X @ nil_b ) )
= nil_a ) ).
% pre_digraph.inner_verts_singleton
thf(fact_1081_pre__digraph_Oinner__verts__def,axiom,
( pre_inner_verts_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,P4: list_b] : ( tl_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ G2 ) @ P4 ) ) ) ) ).
% pre_digraph.inner_verts_def
thf(fact_1082_verts__add__arc__conv,axiom,
! [A2: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A2 ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ t @ A2 ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ t @ A2 ) @ bot_bot_set_a ) ) ) ) ).
% verts_add_arc_conv
thf(fact_1083_arcs__del__leaf,axiom,
! [E: b,V2: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= V2 )
=> ( ( shorte1213025427933718126af_a_b @ t @ V2 )
=> ( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V2 ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ E @ bot_bot_set_b ) ) ) ) ) ) ).
% arcs_del_leaf
thf(fact_1084_connected__verts,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
!= bot_bot_set_b )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( sup_sup_set_a @ ( image_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) @ ( image_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ) ).
% connected_verts
thf(fact_1085_pre__digraph_Overts__del__vert,axiom,
! [G: pre_pr2882871181989701257t_unit,U2: list_a] :
( ( pre_ve1830060048215441954t_unit @ ( pre_de3896127371068354340st_a_b @ G @ U2 ) )
= ( minus_646659088055828811list_a @ ( pre_ve1830060048215441954t_unit @ G ) @ ( insert_list_a @ U2 @ bot_bot_set_list_a ) ) ) ).
% pre_digraph.verts_del_vert
thf(fact_1086_pre__digraph_Overts__del__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ G @ U2 ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ G ) @ ( insert_a @ U2 @ bot_bot_set_a ) ) ) ).
% pre_digraph.verts_del_vert
thf(fact_1087_add__arc__commute,axiom,
! [B2: b,A2: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ B2 ) @ A2 )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ B2 ) ) ).
% add_arc_commute
thf(fact_1088_add__arc__in,axiom,
! [A2: b] :
( ( member_b @ A2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_add_arc_a_b @ t @ A2 )
= t ) ) ).
% add_arc_in
thf(fact_1089_iapath__dist__ends,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( pre_gen_iapath_a_b @ t @ ( verts3_a_b @ t ) @ U2 @ P2 @ V2 )
=> ( U2 != V2 ) ) ).
% iapath_dist_ends
thf(fact_1090_add__add__arc__collapse,axiom,
! [A2: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ A2 )
= ( pre_add_arc_a_b @ t @ A2 ) ) ).
% add_add_arc_collapse
thf(fact_1091_list_Oset__map,axiom,
! [F: a > a,V2: list_a] :
( ( set_a2 @ ( map_a_a @ F @ V2 ) )
= ( image_a_a @ F @ ( set_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1092_list_Oset__map,axiom,
! [F: b > a,V2: list_b] :
( ( set_a2 @ ( map_b_a @ F @ V2 ) )
= ( image_b_a @ F @ ( set_b2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1093_list_Oset__map,axiom,
! [F: a > b,V2: list_a] :
( ( set_b2 @ ( map_a_b @ F @ V2 ) )
= ( image_a_b @ F @ ( set_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1094_list_Oset__map,axiom,
! [F: b > b,V2: list_b] :
( ( set_b2 @ ( map_b_b @ F @ V2 ) )
= ( image_b_b @ F @ ( set_b2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1095_list_Oset__map,axiom,
! [F: a > set_a,V2: list_a] :
( ( set_set_a2 @ ( map_a_set_a @ F @ V2 ) )
= ( image_a_set_a @ F @ ( set_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1096_list_Oset__map,axiom,
! [F: list_a > a,V2: list_list_a] :
( ( set_a2 @ ( map_list_a_a @ F @ V2 ) )
= ( image_list_a_a @ F @ ( set_list_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1097_list_Oset__map,axiom,
! [F: list_a > b,V2: list_list_a] :
( ( set_b2 @ ( map_list_a_b @ F @ V2 ) )
= ( image_list_a_b @ F @ ( set_list_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1098_list_Oset__map,axiom,
! [F: a > list_a,V2: list_a] :
( ( set_list_a2 @ ( map_a_list_a @ F @ V2 ) )
= ( image_a_list_a @ F @ ( set_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1099_list_Oset__map,axiom,
! [F: b > list_a,V2: list_b] :
( ( set_list_a2 @ ( map_b_list_a @ F @ V2 ) )
= ( image_b_list_a @ F @ ( set_b2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1100_list_Oset__map,axiom,
! [F: list_a > set_a,V2: list_list_a] :
( ( set_set_a2 @ ( map_list_a_set_a @ F @ V2 ) )
= ( image_list_a_set_a @ F @ ( set_list_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1101_head__add__arc,axiom,
! [A2: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ t @ A2 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_arc
thf(fact_1102_tail__add__arc,axiom,
! [A2: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ t @ A2 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_arc
thf(fact_1103_add__del__arc__collapse,axiom,
! [A2: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ A2 )
= ( pre_add_arc_a_b @ t @ A2 ) ) ).
% add_del_arc_collapse
thf(fact_1104_arcs__add__arc,axiom,
! [A2: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ t @ A2 ) )
= ( insert_b @ A2 @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% arcs_add_arc
thf(fact_1105_verts__add__arc,axiom,
! [A2: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A2 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A2 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% verts_add_arc
thf(fact_1106_del__add__arc__collapse,axiom,
! [A2: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A2 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A2 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ A2 )
= ( pre_del_arc_a_b @ t @ A2 ) ) ) ) ).
% del_add_arc_collapse
thf(fact_1107_arcs__del__arc,axiom,
! [A2: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ t @ A2 ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ).
% arcs_del_arc
thf(fact_1108_pre__digraph_Ohead__add__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A2: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ G @ A2 ) )
= ( pre_he5236287464308401016t_unit @ G ) ) ).
% pre_digraph.head_add_arc
thf(fact_1109_image__mono,axiom,
! [A3: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( ord_le8200006823705900825t_unit @ A3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A3 ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1110_image__mono,axiom,
! [A3: set_set_a,B3: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A3 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1111_image__mono,axiom,
! [A3: set_list_a,B3: set_list_a,F: list_a > set_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ A3 ) @ ( image_list_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1112_image__mono,axiom,
! [A3: set_a,B3: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1113_image__mono,axiom,
! [A3: set_a,B3: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1114_image__mono,axiom,
! [A3: set_a,B3: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A3 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1115_image__mono,axiom,
! [A3: set_b,B3: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A3 ) @ ( image_b_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1116_image__mono,axiom,
! [A3: set_b,B3: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A3 ) @ ( image_b_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1117_image__subsetI,axiom,
! [A3: set_b,F: b > a,B3: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1118_image__subsetI,axiom,
! [A3: set_a,F: a > a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1119_image__subsetI,axiom,
! [A3: set_b,F: b > b,B3: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( member_b @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1120_image__subsetI,axiom,
! [A3: set_a,F: a > b,B3: set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_b @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1121_image__subsetI,axiom,
! [A3: set_b,F: b > list_a,B3: set_list_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( member_list_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le8861187494160871172list_a @ ( image_b_list_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1122_image__subsetI,axiom,
! [A3: set_b,F: b > set_a,B3: set_set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_b_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1123_image__subsetI,axiom,
! [A3: set_a,F: a > list_a,B3: set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_list_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1124_image__subsetI,axiom,
! [A3: set_a,F: a > set_a,B3: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1125_image__subsetI,axiom,
! [A3: set_list_a,F: list_a > a,B3: set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1126_image__subsetI,axiom,
! [A3: set_set_a,F: set_a > a,B3: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1127_subset__imageE,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A3: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A3 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A3 )
=> ( B3
!= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1128_subset__imageE,axiom,
! [B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A3: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B3
!= ( image_6801035452528096924t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1129_subset__imageE,axiom,
! [B3: set_set_a,F: list_a > set_a,A3: set_list_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A3 ) )
=> ~ ! [C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ A3 )
=> ( B3
!= ( image_list_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1130_subset__imageE,axiom,
! [B3: set_set_a,F: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B3
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1131_subset__imageE,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B3
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1132_subset__imageE,axiom,
! [B3: set_a,F: b > a,A3: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A3 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A3 )
=> ( B3
!= ( image_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1133_subset__imageE,axiom,
! [B3: set_b,F: a > b,A3: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B3
!= ( image_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1134_subset__imageE,axiom,
! [B3: set_b,F: b > b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A3 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A3 )
=> ( B3
!= ( image_b_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1135_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A3: set_set_a,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A3 ) @ B3 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member6939884229742472986t_unit @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1136_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A3: set_pr5411798346947241657t_unit,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A3 ) @ B3 )
= ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A3 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1137_image__subset__iff,axiom,
! [F: a > set_a,A3: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B3 )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1138_image__subset__iff,axiom,
! [F: list_a > set_a,A3: set_list_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ A3 ) @ B3 )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A3 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1139_image__subset__iff,axiom,
! [F: b > a,A3: set_b,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F @ A3 ) @ B3 )
= ( ! [X4: b] :
( ( member_b @ X4 @ A3 )
=> ( member_a @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1140_subset__image__iff,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A3: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A3 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A3 )
& ( B3
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1141_subset__image__iff,axiom,
! [B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A3: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B3
= ( image_6801035452528096924t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1142_subset__image__iff,axiom,
! [B3: set_set_a,F: list_a > set_a,A3: set_list_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A3 ) )
= ( ? [AA: set_list_a] :
( ( ord_le8861187494160871172list_a @ AA @ A3 )
& ( B3
= ( image_list_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1143_subset__image__iff,axiom,
! [B3: set_set_a,F: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B3
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1144_subset__image__iff,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B3
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1145_subset__image__iff,axiom,
! [B3: set_a,F: b > a,A3: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A3 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A3 )
& ( B3
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1146_subset__image__iff,axiom,
! [B3: set_b,F: a > b,A3: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B3
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1147_subset__image__iff,axiom,
! [B3: set_b,F: b > b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A3 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A3 )
& ( B3
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1148_image__Int__subset,axiom,
! [F: a > set_a,A3: set_a,B3: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A3 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1149_image__Int__subset,axiom,
! [F: b > a,A3: set_b,B3: set_b] : ( ord_less_eq_set_a @ ( image_b_a @ F @ ( inf_inf_set_b @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( image_b_a @ F @ A3 ) @ ( image_b_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1150_image__Int__subset,axiom,
! [F: a > a,A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1151_image__Int__subset,axiom,
! [F: a > b,A3: set_a,B3: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_b @ ( image_a_b @ F @ A3 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1152_in__arcs__del__arc__iff,axiom,
! [A2: b,U2: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A2 )
= U2 )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ U2 )
= ( minus_minus_set_b @ ( in_arcs_a_b @ t @ U2 ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A2 )
!= U2 )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ U2 )
= ( in_arcs_a_b @ t @ U2 ) ) ) ) ).
% in_arcs_del_arc_iff
thf(fact_1153_out__arcs__del__arc__iff,axiom,
! [A2: b,U2: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A2 )
= U2 )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ U2 )
= ( minus_minus_set_b @ ( out_arcs_a_b @ t @ U2 ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A2 )
!= U2 )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ U2 )
= ( out_arcs_a_b @ t @ U2 ) ) ) ) ).
% out_arcs_del_arc_iff
thf(fact_1154_in__arcs__add__arc__iff,axiom,
! [A2: b,U2: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A2 )
= U2 )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ U2 )
= ( insert_b @ A2 @ ( in_arcs_a_b @ t @ U2 ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A2 )
!= U2 )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ U2 )
= ( in_arcs_a_b @ t @ U2 ) ) ) ) ).
% in_arcs_add_arc_iff
thf(fact_1155_scc__disj,axiom,
! [C: pre_pr7278220950009878019t_unit,D2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D2 @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( C != D2 )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D2 ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_1156_not__elem__no__in__arcs,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( in_arcs_a_b @ t @ V2 )
= bot_bot_set_b ) ) ).
% not_elem_no_in_arcs
thf(fact_1157_not__elem__no__out__arcs,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( out_arcs_a_b @ t @ V2 )
= bot_bot_set_b ) ) ).
% not_elem_no_out_arcs
thf(fact_1158_in__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,D2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D2 @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D2 ) )
=> ( C = D2 ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_1159_sccs__verts__conv,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% sccs_verts_conv
thf(fact_1160_in__sccs__vertsI__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% in_sccs_vertsI_sccs
thf(fact_1161_sccs__verts__conv__scc__of,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ t ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_conv_scc_of
thf(fact_1162_sccs__conv__sccs__verts,axiom,
( ( digraph_pre_sccs_a_b @ t )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ t ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% sccs_conv_sccs_verts
thf(fact_1163_in__sccs__verts__conv,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_sccs_verts_conv
thf(fact_1164_in__verts__sccsD__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_verts_sccsD_sccs
thf(fact_1165_out__arcs__add__arc__iff,axiom,
! [A2: b,U2: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A2 )
= U2 )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ U2 )
= ( insert_b @ A2 @ ( out_arcs_a_b @ t @ U2 ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A2 )
!= U2 )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A2 ) @ U2 )
= ( out_arcs_a_b @ t @ U2 ) ) ) ) ).
% out_arcs_add_arc_iff
thf(fact_1166_leaf__def,axiom,
! [V2: a] :
( ( shorte1213025427933718126af_a_b @ t @ V2 )
= ( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ( out_arcs_a_b @ t @ V2 )
= bot_bot_set_b ) ) ) ).
% leaf_def
thf(fact_1167_arcs__del__vert2,axiom,
! [V2: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( in_arcs_a_b @ t @ V2 ) ) @ ( out_arcs_a_b @ t @ V2 ) ) ) ).
% arcs_del_vert2
thf(fact_1168_to__list__tree__union__verts__eq,axiom,
( ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% to_list_tree_union_verts_eq
thf(fact_1169_reachable__vwalk__conv,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
= ( ? [P4: list_a] :
( ( vertex_vwalk_a_b @ P4 @ t )
& ( ( hd_a @ P4 )
= U2 )
& ( ( last_a @ P4 )
= V2 ) ) ) ) ).
% reachable_vwalk_conv
thf(fact_1170_awalk__imp__vwalk,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( vertex_vwalk_a_b @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) @ t ) ) ).
% awalk_imp_vwalk
thf(fact_1171_unvis__insert,axiom,
! [U2: a,X: a,U: set_a] :
( ( graph_2016941059203891550ts_a_b @ t @ U2 @ ( insert_a @ X @ U ) )
= ( minus_minus_set_a @ ( graph_2016941059203891550ts_a_b @ t @ U2 @ U ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% unvis_insert
thf(fact_1172_induce__eq__iff__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ t )
=> ( ( digrap7873285959652527175ph_a_b @ t @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ).
% induce_eq_iff_induced
thf(fact_1173_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ t @ t ).
% induced_subgraph_refl
thf(fact_1174_in__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( digrap5251062021860773499ph_a_b @ C @ t ) ) ).
% in_sccs_imp_induced
thf(fact_1175_disj__unvis__vis,axiom,
! [U2: a,U: set_a] :
( ( inf_inf_set_a @ ( graph_2016941059203891550ts_a_b @ t @ U2 @ U ) @ U )
= bot_bot_set_a ) ).
% disj_unvis_vis
thf(fact_1176_induced__induce,axiom,
! [Vs: set_a] :
( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ t @ Vs ) @ t ) ) ).
% induced_induce
thf(fact_1177_induced__subgraph__altdef,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ t )
= ( ( digraph_subgraph_a_b @ H @ t )
& ! [H2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H2 @ t )
=> ( ( ( pre_ve642382030648772252t_unit @ H2 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H2 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) ) ) ) ).
% induced_subgraph_altdef
thf(fact_1178_strongly__connected__spanning__imp__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ t )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ t ) ) ) ).
% strongly_connected_spanning_imp_strongly_connected
thf(fact_1179_subgraph__refl,axiom,
digraph_subgraph_a_b @ t @ t ).
% subgraph_refl
thf(fact_1180_subgraph__awalk__imp__awalk,axiom,
! [H: pre_pr7278220950009878019t_unit,U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ H @ U2 @ P2 @ V2 )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 ) ) ) ).
% subgraph_awalk_imp_awalk
thf(fact_1181_reachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U2: a,V2: a] :
( ( reachable_a_b @ H @ U2 @ V2 )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( reachable_a_b @ t @ U2 @ V2 ) ) ) ).
% reachable_mono
thf(fact_1182_subgraph__apath__imp__apath,axiom,
! [H: pre_pr7278220950009878019t_unit,U2: a,P2: list_b,V2: a] :
( ( arc_pre_apath_a_b @ H @ U2 @ P2 @ V2 )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( arc_pre_apath_a_b @ t @ U2 @ P2 @ V2 ) ) ) ).
% subgraph_apath_imp_apath
thf(fact_1183_subgraph__del__arc,axiom,
! [A2: b] : ( digraph_subgraph_a_b @ ( pre_del_arc_a_b @ t @ A2 ) @ t ) ).
% subgraph_del_arc
thf(fact_1184_subgraph__del__vert,axiom,
! [U2: a] : ( digraph_subgraph_a_b @ ( pre_del_vert_a_b @ t @ U2 ) @ t ) ).
% subgraph_del_vert
thf(fact_1185_strongly__connected__eq__iff,axiom,
( ( digrap8691851296217657702ed_a_b @ t )
= ( ( digraph_pre_sccs_a_b @ t )
= ( insert6864688055023459379t_unit @ t @ bot_bo1839476491465656141t_unit ) ) ) ).
% strongly_connected_eq_iff
thf(fact_1186_subgraph__cycle,axiom,
! [H: pre_pr7278220950009878019t_unit,P2: list_b] :
( ( digraph_subgraph_a_b @ H @ t )
=> ( ( arc_pre_cycle_a_b @ H @ P2 )
=> ( arc_pre_cycle_a_b @ t @ P2 ) ) ) ).
% subgraph_cycle
thf(fact_1187_subgraph__induce__subgraphI,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( digraph_subgraph_a_b @ ( digrap7873285959652527175ph_a_b @ t @ V3 ) @ t ) ) ).
% subgraph_induce_subgraphI
thf(fact_1188_strongly__connected__imp__induce__subgraph__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ t )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ ( digrap7873285959652527175ph_a_b @ t @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ) ).
% strongly_connected_imp_induce_subgraph_strongly_connected
thf(fact_1189_symmetric__connected__imp__strongly__connected,axiom,
( ( symmetric_a_b @ t )
=> ( ( digrap8783888973171253482ed_a_b @ t )
=> ( digrap8691851296217657702ed_a_b @ t ) ) ) ).
% symmetric_connected_imp_strongly_connected
thf(fact_1190_in__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ t )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ t )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% in_sccsE
thf(fact_1191_in__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ t )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ t )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) ) ) ) ) ).
% in_sccsI
thf(fact_1192_branch__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ C2 ) )
=> ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ) ).
% branch_in_supergraph
thf(fact_1193_cas__takeI,axiom,
! [U2: a,P2: list_b,V2: a,N: nat,V4: a] :
( ( arc_pre_cas_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ ( take_b @ N @ P2 ) ) )
= V4 )
=> ( arc_pre_cas_a_b @ t @ U2 @ ( take_b @ N @ P2 ) @ V4 ) ) ) ).
% cas_takeI
thf(fact_1194_ends__del__vert,axiom,
! [U2: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ t @ U2 ) )
= ( arc_to_ends_a_b @ t ) ) ).
% ends_del_vert
thf(fact_1195_branch__in__verts,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ t ) )
=> ( member_a @ X @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% branch_in_verts
thf(fact_1196_nomulti_Ono__multi__arcs,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E1 )
= ( arc_to_ends_a_b @ t @ E22 ) )
=> ( E1 = E22 ) ) ) ) ).
% nomulti.no_multi_arcs
thf(fact_1197_is__chain__def,axiom,
( ( graph_3890552050688490787in_a_b @ t )
= ( ( graph_4596510882073158607ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain_def
thf(fact_1198_awalk__induct__raw,axiom,
! [U2: a,P2: list_b,V2: a,P: a > list_b > a > $o] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ! [W1: a] :
( ( member_a @ W1 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ W1 @ nil_b @ W1 ) )
=> ( ! [W1: a,W22: a,E2: b,Es2: list_b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E2 )
= ( product_Pair_a_a @ W1 @ W22 ) )
=> ( ( P @ W22 @ Es2 @ V2 )
=> ( P @ W1 @ ( cons_b @ E2 @ Es2 ) @ V2 ) ) ) )
=> ( P @ U2 @ P2 @ V2 ) ) ) ) ).
% awalk_induct_raw
thf(fact_1199_last__branch__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_1747835947655717337ts_a_b @ t ) )
=> ! [Z3: a] :
( ( ( reachable_a_b @ t @ X @ Z3 )
& ( Z3 != X ) )
=> ~ ( member_a @ Z3 @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ) ).
% last_branch_alt
thf(fact_1200_awalk__ConsI,axiom,
! [V2: a,Es: list_b,W: a,E: b,U2: a] :
( ( arc_pre_awalk_a_b @ t @ V2 @ Es @ W )
=> ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E )
= ( product_Pair_a_a @ U2 @ V2 ) )
=> ( arc_pre_awalk_a_b @ t @ U2 @ ( cons_b @ E @ Es ) @ W ) ) ) ) ).
% awalk_ConsI
thf(fact_1201_last__branch__is__branch,axiom,
! [Y: a] :
( ( member_a @ Y @ ( graph_1747835947655717337ts_a_b @ t ) )
=> ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ).
% last_branch_is_branch
thf(fact_1202_reachable__arc__trans,axiom,
! [U2: a,V2: a,E: b,W: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ V2 @ W ) )
=> ( reachable_a_b @ t @ U2 @ W ) ) ) ).
% reachable_arc_trans
thf(fact_1203_arcE,axiom,
! [E: b,U2: a,V2: a] :
( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ U2 @ V2 ) )
=> ~ ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U2 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E )
!= V2 ) ) ) ) ).
% arcE
thf(fact_1204_no__back__before__aux,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
=> ( ( iKKBZ_4622586873178280511rm_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ) ) ) ).
% no_back_before_aux
thf(fact_1205_before__def,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
= ( ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 )
& ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 )
& ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
& ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_def
thf(fact_1206_cas_Ocases,axiom,
! [X: produc7945266988514096265st_b_a] :
( ! [U5: a,V: a] :
( X
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ nil_b @ V ) ) )
=> ~ ! [U5: a,E2: b,Es2: list_b,V: a] :
( X
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es2 ) @ V ) ) ) ) ).
% cas.cases
thf(fact_1207_loopfree_Oadj__not__same,axiom,
! [A2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( arcs_ends_a_b @ t ) ) ).
% loopfree.adj_not_same
thf(fact_1208_adj__in__verts_I2_J,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(2)
thf(fact_1209_adj__in__verts_I1_J,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(1)
thf(fact_1210_awalk__dom__if__uneq,axiom,
! [U2: a,V2: a,P2: list_b] :
( ( U2 != V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ? [X2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ V2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% awalk_dom_if_uneq
thf(fact_1211_adj__reachable__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ B2 @ C )
=> ( reachable_a_b @ t @ A2 @ C ) ) ) ).
% adj_reachable_trans
thf(fact_1212_reachable__adj__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( reachable_a_b @ t @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B2 @ C ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ A2 @ C ) ) ) ).
% reachable_adj_trans
thf(fact_1213_reachable__via__child__impl__same,axiom,
! [X: a,V2: a,Y: a,U2: a] :
( ( reachable_a_b @ t @ X @ V2 )
=> ( ( reachable_a_b @ t @ Y @ V2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ X ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( X = Y ) ) ) ) ) ).
% reachable_via_child_impl_same
thf(fact_1214_adj__mono,axiom,
! [U2: a,V2: a,H: pre_pr7278220950009878019t_unit] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ H ) )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% adj_mono
thf(fact_1215_reachable__induct,axiom,
! [U2: a,V2: a,P: a > $o] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ U2 ) )
=> ( ! [X2: a,Y2: a] :
( ( reachable_a_b @ t @ U2 @ X2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( P @ X2 )
=> ( P @ Y2 ) ) ) )
=> ( P @ V2 ) ) ) ) ).
% reachable_induct
thf(fact_1216_converse__reachable__induct,axiom,
! [U2: a,V2: a,P: a > $o] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ V2 ) )
=> ( ! [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ Y2 @ V2 )
=> ( ( P @ Y2 )
=> ( P @ X2 ) ) ) )
=> ( P @ U2 ) ) ) ) ).
% converse_reachable_induct
thf(fact_1217_converse__reachable__cases,axiom,
! [U2: a,V2: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( ( U2 = V2 )
=> ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) )
=> ~ ! [W2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ W2 ) @ ( arcs_ends_a_b @ t ) )
=> ~ ( reachable_a_b @ t @ W2 @ V2 ) ) ) ) ).
% converse_reachable_cases
thf(fact_1218_before__ArcI,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ S1 ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% before_ArcI
thf(fact_1219_dominatesI,axiom,
! [A2: b,U2: a,V2: a] :
( ( ( arc_to_ends_a_b @ t @ A2 )
= ( product_Pair_a_a @ U2 @ V2 ) )
=> ( ( member_b @ A2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% dominatesI
thf(fact_1220_dominates__induce__ss,axiom,
! [U2: a,V2: a,S: set_a,T: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) ) )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T ) ) ) ) ) ).
% dominates_induce_ss
thf(fact_1221_dominated__notin__awalk,axiom,
! [U2: a,V2: a,R2: a,P2: list_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ P2 @ U2 )
=> ~ ( member_a @ V2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P2 ) ) ) ) ) ).
% dominated_notin_awalk
thf(fact_1222_awalk__verts__dom__if__uneq,axiom,
! [U2: a,V2: a,P2: list_b] :
( ( U2 != V2 )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ? [X2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
& ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) ) ) ) ) ) ).
% awalk_verts_dom_if_uneq
thf(fact_1223_unique__arc_I2_J,axiom,
! [U2: a,V2: a] :
( ~ ? [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= U2 )
& ( ( pre_he5236287464308401016t_unit @ t @ E2 )
= V2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) ) ) ).
% unique_arc(2)
thf(fact_1224_unique__arc_I1_J,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
=> ? [X2: b] :
( ( member_b @ X2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ X2 )
= U2 )
& ( ( pre_he5236287464308401016t_unit @ t @ X2 )
= V2 )
& ! [Y4: b] :
( ( ( member_b @ Y4 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ Y4 )
= U2 )
& ( ( pre_he5236287464308401016t_unit @ t @ Y4 )
= V2 ) )
=> ( Y4 = X2 ) ) ) ) ).
% unique_arc(1)
thf(fact_1225_in__arcs__imp__in__arcs__ends,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% in_arcs_imp_in_arcs_ends
thf(fact_1226_forward__arcs_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( cons_a @ V @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va ) ) ) ) ) ).
% forward_arcs.elims(3)
thf(fact_1227_forward__arcs_Osimps_I3_J,axiom,
! [X: a,V2: a,Va2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ ( cons_a @ V2 @ Va2 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( cons_a @ V2 @ Va2 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V2 @ Va2 ) ) ) ) ).
% forward_arcs.simps(3)
thf(fact_1228_forward__arc__to__head_H,axiom,
! [Ys: list_a,X: a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( Y
= ( hd_a @ Ys ) ) ) ) ) ) ).
% forward_arc_to_head'
thf(fact_1229_no__arc__fst__if__no__back,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ Xs ) )
=> ( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% no_arc_fst_if_no_back
thf(fact_1230_no__back__arcs_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
=> ~ ! [X2: a,Xs2: list_a] :
( ( X
= ( cons_a @ X2 @ Xs2 ) )
=> ( ~ ? [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ).
% no_back_arcs.elims(3)
thf(fact_1231_no__back__arcs_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X @ Xs ) )
= ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% no_back_arcs.simps(2)
thf(fact_1232_before__arc__to__hd,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% before_arc_to_hd
thf(fact_1233_vwalk__wf__digraph__consI,axiom,
! [P2: list_a,A2: a] :
( ( vertex_vwalk_a_b @ P2 @ t )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ ( hd_a @ P2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A2 @ P2 ) @ t ) ) ) ).
% vwalk_wf_digraph_consI
thf(fact_1234_move__mid__forward__if__noarc,axiom,
! [As: list_a,U: list_a,Bs: list_a,Cs2: list_a] :
( ( As != nil_a )
=> ( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ U ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa3 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ Bs @ Cs2 ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U @ Cs2 ) ) ) ) ) ) ) ).
% move_mid_forward_if_noarc
thf(fact_1235_arc__to__lst__if__forward,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs ) ) )
=> ( ( Xs
= ( cons_a @ Y @ Ys ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% arc_to_lst_if_forward
thf(fact_1236_loopfree_OvpathI__arc,axiom,
! [A2: a,B2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vpath_a_b @ ( cons_a @ A2 @ ( cons_a @ B2 @ nil_a ) ) @ t ) ) ).
% loopfree.vpathI_arc
thf(fact_1237_no__back__arc__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% no_back_arc_if_fwd_dstct
thf(fact_1238_forward__arcs_Oelims_I2_J,axiom,
! [X: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
=> ( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ~ ( ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( cons_a @ V @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% forward_arcs.elims(2)
thf(fact_1239_forward__arcs_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
= Y )
=> ( ( ( X = nil_a )
=> ~ Y )
=> ( ( ? [X2: a] :
( X
= ( cons_a @ X2 @ nil_a ) )
=> ~ Y )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( Y
= ( ~ ( ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ V @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% forward_arcs.elims(1)
thf(fact_1240_forward__app,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ ( hd_a @ S2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ).
% forward_app
thf(fact_1241_no__back__arcs_Oelims_I2_J,axiom,
! [X: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
=> ( ( X != nil_a )
=> ~ ! [X2: a,Xs2: list_a] :
( ( X
= ( cons_a @ X2 @ Xs2 ) )
=> ~ ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ).
% no_back_arcs.elims(2)
thf(fact_1242_no__back__arcs_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
= Y )
=> ( ( ( X = nil_a )
=> ~ Y )
=> ~ ! [X2: a,Xs2: list_a] :
( ( X
= ( cons_a @ X2 @ Xs2 ) )
=> ( Y
= ( ~ ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ) ) ).
% no_back_arcs.elims(1)
thf(fact_1243_forward__app_H,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
=> ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ S1 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ) ).
% forward_app'
thf(fact_1244_move__mid__backward__if__noarc_H,axiom,
! [U: list_a,V3: list_a,As: list_a,Bs: list_a,Cs2: list_a] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ U ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ V3 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ Bs @ ( append_a @ V3 @ Cs2 ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ V3 @ ( append_a @ Bs @ Cs2 ) ) ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc'
thf(fact_1245_forward__arc__to__head,axiom,
! [Ys: list_a,Xs: list_a,X: a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( Y
= ( hd_a @ Ys ) ) ) ) ) ) ) ).
% forward_arc_to_head
thf(fact_1246_reachable__adjI,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ U2 @ V2 ) ) ).
% reachable_adjI
thf(fact_1247_vwalk__Cons__Cons,axiom,
! [U2: a,V2: a,Ws: list_a] :
( ( vertex_vwalk_a_b @ ( cons_a @ U2 @ ( cons_a @ V2 @ Ws ) ) @ t )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( arcs_ends_a_b @ t ) )
& ( vertex_vwalk_a_b @ ( cons_a @ V2 @ Ws ) @ t ) ) ) ).
% vwalk_Cons_Cons
thf(fact_1248_before2__def,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_1040310085189658461e2_a_b @ t @ S1 @ S2 )
= ( ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 )
& ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 )
& ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
& ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( arcs_ends_a_b @ t ) ) ) )
& ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( minus_minus_set_a @ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( set_a2 @ S1 ) ) @ ( set_a2 @ S2 ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before2_def
thf(fact_1249_awalk__cyc__decompE,axiom,
! [P2: list_b,Q: list_b,R2: list_b,S4: list_b,U2: a,V2: a] :
( ( ( arc_wf4740610840468824943mp_a_b @ t @ P2 )
= ( produc305491333965050169list_b @ Q @ ( produc1564554178308465111list_b @ R2 @ S4 ) ) )
=> ( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
=> ~ ( ( P2
= ( append_b @ Q @ ( append_b @ R2 @ S4 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ Q ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ Q @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S4 @ V2 ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R2 ) ) ) ) ) ) ) ).
% awalk_cyc_decompE
thf(fact_1250_step__awalk__to__apath,axiom,
! [U2: a,P2: list_b,V2: a,Q: list_b,R2: list_b,S4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P2 )
= ( produc305491333965050169list_b @ Q @ ( produc1564554178308465111list_b @ R2 @ S4 ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
=> ( ( arc_wf446166946845163101th_a_b @ t @ P2 )
= ( arc_wf446166946845163101th_a_b @ t @ ( append_b @ Q @ S4 ) ) ) ) ) ) ).
% step_awalk_to_apath
thf(fact_1251_is__awalk__cyc__decomp_Ocases,axiom,
! [X: produc272433356463431595list_b] :
~ ! [P5: list_b,Q2: list_b,R: list_b,S3: list_b] :
( X
!= ( produc7106373121284446491list_b @ P5 @ ( produc305491333965050169list_b @ Q2 @ ( produc1564554178308465111list_b @ R @ S3 ) ) ) ) ).
% is_awalk_cyc_decomp.cases
thf(fact_1252_to__list__tree__dom__iff,axiom,
! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ nil_a ) @ ( cons_a @ Y @ nil_a ) ) @ ( arcs_ends_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) ) ) ) ).
% to_list_tree_dom_iff
thf(fact_1253_awalk__to__apath__induct,axiom,
! [U2: a,P2: list_b,V2: a,P: list_b > $o] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ! [P5: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P5 @ V2 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P5 ) )
=> ( P @ P5 ) ) )
=> ( ! [P5: list_b,Q2: list_b,R: list_b,S3: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P5 @ V2 )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P5 )
= ( produc305491333965050169list_b @ Q2 @ ( produc1564554178308465111list_b @ R @ S3 ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P5 ) )
=> ( ( P @ ( append_b @ Q2 @ S3 ) )
=> ( P @ P5 ) ) ) ) )
=> ( P @ P2 ) ) ) ) ).
% awalk_to_apath_induct
thf(fact_1254_awalk__cyc__decomp__has__prop,axiom,
! [U2: a,P2: list_b,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P2 @ V2 )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U2 @ P2 ) )
=> ( arc_wf7293661141070756729mp_a_b @ t @ P2 @ ( arc_wf4740610840468824943mp_a_b @ t @ P2 ) ) ) ) ).
% awalk_cyc_decomp_has_prop
thf(fact_1255_hd__reachable1__from__outside,axiom,
! [X: a,Y: a,Ys: list_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ? [X2: a] : ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside
thf(fact_1256_reachable1__not__reverse,axiom,
! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ).
% reachable1_not_reverse
thf(fact_1257_reachable1__from__outside__dom,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ? [X7: a,X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ~ ( member_a @ X7 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X7 @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% reachable1_from_outside_dom
thf(fact_1258_reachable1__in__verts_I2_J,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(2)
thf(fact_1259_reachable1__in__verts_I1_J,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(1)
thf(fact_1260_reachable1__reachable__trans,axiom,
! [U2: a,V2: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( reachable_a_b @ t @ V2 @ W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_reachable_trans
thf(fact_1261_reachable__reachable1__trans,axiom,
! [U2: a,V2: a,W: a] :
( ( reachable_a_b @ t @ U2 @ V2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_reachable1_trans
thf(fact_1262_reachable1__awalk,axiom,
! [U2: a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
= ( ? [P4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U2 @ P4 @ V2 )
& ( P4 != nil_b ) ) ) ) ).
% reachable1_awalk
thf(fact_1263_reachable1__awalkI,axiom,
! [V2: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ V2 @ P2 @ W )
=> ( ( P2 != nil_b )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_awalkI
thf(fact_1264_reachable1__append__old__if__arc,axiom,
! [Xs: list_a,Ys: list_a,Z: a,Y: a] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ( member_a @ Z @ ( set_a2 @ Xs ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arc
thf(fact_1265_hd__reachable1__from__outside_H,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ? [X2: a] : ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside'
thf(fact_1266_no__back__reach1__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct
thf(fact_1267_not__reachable1__append__if__not__old,axiom,
! [U: list_a,B2: list_a,X: list_a] :
( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ U ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ B2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa3 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U ) @ ( set_a2 @ X ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ X )
=> ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ X ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ B2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ U ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( append_a @ X @ B2 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% not_reachable1_append_if_not_old
thf(fact_1268_reachable1__append__old__if__arcU,axiom,
! [Xs: list_a,Ys: list_a,U: list_a,Z: a,Y: a] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U ) @ ( set_a2 @ Xs ) )
= bot_bot_set_a )
=> ( ( member_a @ Z @ ( set_a2 @ U ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arcU
thf(fact_1269_hd__reach1__y__if__nfwd__app__fwd,axiom,
! [Y: a,Xs: list_a,Ys: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ Xs ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( hd_a @ ( rev_a @ ( cons_a @ Y @ ( append_a @ Ys @ Xs ) ) ) ) @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% hd_reach1_y_if_nfwd_app_fwd
thf(fact_1270_reachable__neq__reachable1,axiom,
! [V2: a,W: a] :
( ( reachable_a_b @ t @ V2 @ W )
=> ( ( V2 != W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_neq_reachable1
thf(fact_1271_reachable1__reachable,axiom,
! [V2: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V2 @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( reachable_a_b @ t @ V2 @ W ) ) ).
% reachable1_reachable
thf(fact_1272_no__back__reach1__if__fwd__dstct__bs,axiom,
! [As: list_a,Bs: list_list_a,V3: list_a,Cs2: list_a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V3 @ Cs2 ) ) ) )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V3 @ Cs2 ) ) ) )
=> ( ( member_list_a @ Xs @ ( set_list_a2 @ Bs ) )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ V3 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct_bs
thf(fact_1273_verts__finite__imp__arcs__finite,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% verts_finite_imp_arcs_finite
thf(fact_1274_in__arcs__finite,axiom,
! [V2: a] :
( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( finite_finite_b @ ( in_arcs_a_b @ t @ V2 ) ) ) ).
% in_arcs_finite
thf(fact_1275_ex__leaf,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( shorte1213025427933718126af_a_b @ t @ X2 ) ) ) ).
% ex_leaf
thf(fact_1276_finite__branch__impl__last__branch,axiom,
! [X: a,R2: a] :
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ? [X5: a] :
( ( member_a @ X5 @ ( graph_4596510882073158607ts_a_b @ t ) )
& ( reachable_a_b @ t @ X @ X5 ) )
=> ( ( shorte3810566709427824352ee_a_b @ t @ R2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( graph_1747835947655717337ts_a_b @ t ) )
& ( reachable_a_b @ t @ X @ X2 ) ) ) ) ) ).
% finite_branch_impl_last_branch
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X: list_b,Y: list_b] :
( ( if_list_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X: list_b,Y: list_b] :
( ( if_list_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ x @ ( append_a @ xs @ ( cons_a @ x @ nil_a ) ) ) ) ).
%------------------------------------------------------------------------------